Content-Type: multipart/mixed; boundary="-------------0007281740629" This is a multi-part message in MIME format. ---------------0007281740629 Content-Type: text/plain; name="00-307.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-307.keywords" aperiodic tiling, quasicrystals, substitution tiling ---------------0007281740629 Content-Type: application/postscript; name="cech10.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="cech10.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: cech10.dvi %%Pages: 32 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMBX10 CMR10 CMR7 CMMI7 CMEX10 CMMI10 MSBM10 CMSY10 %%+ CMSY7 CMTI10 CMMI5 MSBM7 CMTT10 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -o cech10.ps cech10 %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2000.07.28:1729 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 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B5FBF780CF8754FA7422C3F57E62CCEEAD9AA3470000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /parenleftbig put dup 162 /parenrightbig put dup 163 /bracketleftbig put dup 164 /bracketrightbig put dup 165 /floorleftbig put dup 166 /floorrightbig put dup 167 /ceilingleftbig put dup 168 /ceilingrightbig put dup 169 /braceleftbig put dup 170 /bracerightbig put dup 173 /angbracketleftbig put dup 174 /angbracketrightbig put dup 175 /vextendsingle put dup 176 /vextenddouble put dup 177 /slashbig put dup 178 /backslashbig put dup 179 /parenleftBig put dup 180 /parenrightBig put dup 181 /parenleftbigg put dup 182 /parenrightbigg put dup 183 /bracketleftbigg put dup 184 /bracketrightbigg put dup 185 /floorleftbigg put dup 186 /floorrightbigg put dup 187 /ceilingleftbigg put dup 188 /ceilingrightbigg put dup 189 /braceleftbigg put dup 190 /bracerightbigg put dup 191 /angbracketleftbigg put dup 192 /angbracketrightbigg put dup 193 /slashbigg put dup 194 /backslashbigg put dup 195 /parenleftBigg put dup 196 /arrowdblbt put dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 4 /floorleftbig put dup 5 /floorrightbig put dup 6 /ceilingleftbig put dup 7 /ceilingrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 14 /slashbig put dup 15 /backslashbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 22 /floorleftbigg put dup 23 /floorrightbigg put dup 24 /ceilingleftbigg put dup 25 /ceilingrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 30 /slashbigg put dup 31 /backslashbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 36 /floorleftBigg put dup 37 /floorrightBigg put dup 38 /ceilingleftBigg put dup 39 /ceilingrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 42 /angbracketleftBigg put dup 43 /angbracketrightBigg put dup 44 /slashBigg put dup 45 /backslashBigg put dup 46 /slashBig put dup 47 /backslashBig put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 54 /bracketleftex put dup 55 /bracketrightex put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 63 /arrowvertex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 70 /unionsqtext put dup 71 /unionsqdisplay put dup 72 /contintegraltext put dup 73 /contintegraldisplay put dup 74 /circledottext put dup 75 /circledotdisplay put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 85 /unionmultitext put dup 86 /logicalandtext put dup 87 /logicalortext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 93 /unionmultidisplay put dup 94 /logicalanddisplay put dup 95 /logicalordisplay put dup 96 /coproducttext put dup 97 /coproductdisplay put dup 98 /hatwide put dup 99 /hatwider put dup 100 /hatwidest put dup 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec 80347982AB3942D930E069A70D0D48311D725E830D1C76FBA12E12486E989C9874C2B527F092 5722787027F44470D484262C360CDFDDDF3657533A57BB16F73048BFBBFCB73A650484015441 FDC837ADD94AC8FBD2022E3EC8F115D4B4BB7B7F15388F22CC6198EFE768BD9FCEB3446EE4A8 DC27D6CD152485384EF5F59381FFDA43F2D20C8FB08AA27AB2015B774DB10DACFDCD33E60F17 8C461553146AB427BDD7DA12534BA078AD3D780414930DA4F8D58ABEFD45DB119B10EB409DD8 97923C6E705479464A4B310B58348C4B42393988FEF4925CF984423AAF65FEA9F0E64629953B CF50B919D968D99BD185F83112B2759CC411764E9BDE677F57C5EE5AC555448D5F81A16259DE 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Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /lessornotequal put dup 162 /greaterornotequal put dup 163 /notlessequal put dup 164 /notgreaterequal put dup 165 /notless put dup 166 /notgreater put dup 167 /notprecedes put dup 168 /notfollows put dup 169 /lessornotdbleql put dup 170 /greaterornotdbleql put dup 173 /notlessorslnteql put dup 174 /notgreaterorslnteql put dup 175 /lessnotequal put dup 176 /greaternotequal put dup 177 /notprecedesoreql put dup 178 /notfollowsoreql put dup 179 /precedeornoteqvlnt put dup 180 /followornoteqvlnt put dup 181 /lessornotsimilar put dup 182 /greaterornotsimilar put dup 183 /notlessdblequal put dup 184 /notgreaterdblequal put dup 185 /precedenotslnteql put dup 186 /follownotslnteql put dup 187 /precedenotdbleqv put dup 188 /follownotdbleqv put dup 189 /lessnotdblequal put dup 190 /greaternotdblequal put dup 191 /notsimilar put dup 192 /notapproxequal put dup 193 /upslope put dup 194 /downslope put dup 195 /notsubsetoreql put dup 196 /epsiloninv put dup 0 /lessornotequal put dup 1 /greaterornotequal put dup 2 /notlessequal put dup 3 /notgreaterequal put dup 4 /notless put dup 5 /notgreater put dup 6 /notprecedes put dup 7 /notfollows put dup 8 /lessornotdbleql put dup 9 /greaterornotdbleql put dup 10 /notlessorslnteql put dup 11 /notgreaterorslnteql put dup 12 /lessnotequal put dup 13 /greaternotequal put dup 14 /notprecedesoreql put dup 15 /notfollowsoreql put dup 16 /precedeornoteqvlnt put dup 17 /followornoteqvlnt put dup 18 /lessornotsimilar put dup 19 /greaterornotsimilar put dup 20 /notlessdblequal put dup 21 /notgreaterdblequal put dup 22 /precedenotslnteql put dup 23 /follownotslnteql put dup 24 /precedenotdbleqv put dup 25 /follownotdbleqv put dup 26 /lessnotdblequal put dup 27 /greaternotdblequal put dup 28 /notsimilar put dup 29 /notapproxequal put dup 30 /upslope put dup 31 /downslope put dup 32 /notsubsetoreql put dup 33 /notsupersetoreql put dup 34 /notsubsetordbleql put dup 35 /notsupersetordbleql put dup 36 /subsetornotdbleql put dup 37 /supersetornotdbleql put dup 38 /subsetornoteql put dup 39 /supersetornoteql put dup 40 /subsetnoteql put dup 41 /supersetnoteql put dup 42 /notsubseteql put dup 43 /notsuperseteql put dup 44 /notparallel put dup 45 /notbar put dup 46 /notshortbar put dup 47 /notshortparallel put dup 48 /notturnstile put dup 49 /notforces put dup 50 /notsatisfies put dup 51 /notforcesextra put dup 52 /nottriangeqlright put dup 53 /nottriangeqlleft put dup 54 /nottriangleleft put dup 55 /nottriangleright put dup 56 /notarrowleft put dup 57 /notarrowright put dup 58 /notdblarrowleft put dup 59 /notdblarrowright put dup 60 /notdblarrowboth put dup 61 /notarrowboth put dup 62 /dividemultiply put dup 63 /emptyset put dup 64 /notexistential put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /hatwide put dup 92 /hatwider put dup 93 /tildewide put dup 94 /tildewider put dup 96 /Finv put dup 97 /Gmir put dup 102 /Omegainv put dup 103 /eth put dup 104 /equalorsimilar put dup 105 /beth put dup 106 /gimel put dup 107 /daleth put dup 108 /lessdot put dup 109 /greaterdot put dup 110 /multicloseleft put dup 111 /multicloseright put dup 112 /barshort put dup 113 /parallelshort put dup 114 /integerdivide put dup 115 /similar put dup 116 /approxequal put dup 117 /approxorequal put dup 118 /followsorequal put dup 119 /precedesorequal put dup 120 /archleftdown put dup 121 /archrightdown put dup 122 /Digamma put dup 123 /kappa put dup 124 /k put dup 125 /planckover2pi put dup 126 /planckover2pi1 put dup 127 /epsiloninv put dup 128 /notsubsetoreql put dup 160 /space put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /sterling put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /lessornotequal put dup 162 /greaterornotequal put dup 163 /notlessequal put dup 164 /notgreaterequal put dup 165 /notless put dup 166 /notgreater put dup 167 /notprecedes put dup 168 /notfollows put dup 169 /lessornotdbleql put dup 170 /greaterornotdbleql put dup 173 /notlessorslnteql put dup 174 /notgreaterorslnteql put dup 175 /lessnotequal put dup 176 /greaternotequal put dup 177 /notprecedesoreql put dup 178 /notfollowsoreql put dup 179 /precedeornoteqvlnt put dup 180 /followornoteqvlnt put dup 181 /lessornotsimilar put dup 182 /greaterornotsimilar put dup 183 /notlessdblequal put dup 184 /notgreaterdblequal put dup 185 /precedenotslnteql put dup 186 /follownotslnteql put dup 187 /precedenotdbleqv put dup 188 /follownotdbleqv put dup 189 /lessnotdblequal put dup 190 /greaternotdblequal put dup 191 /notsimilar put dup 192 /notapproxequal put dup 193 /upslope put dup 194 /downslope put dup 195 /notsubsetoreql put dup 196 /epsiloninv put dup 0 /lessornotequal put dup 1 /greaterornotequal put dup 2 /notlessequal put dup 3 /notgreaterequal put dup 4 /notless put dup 5 /notgreater put dup 6 /notprecedes put dup 7 /notfollows put dup 8 /lessornotdbleql put dup 9 /greaterornotdbleql put dup 10 /notlessorslnteql put dup 11 /notgreaterorslnteql put dup 12 /lessnotequal put dup 13 /greaternotequal put dup 14 /notprecedesoreql put dup 15 /notfollowsoreql put dup 16 /precedeornoteqvlnt put dup 17 /followornoteqvlnt put dup 18 /lessornotsimilar put dup 19 /greaterornotsimilar put dup 20 /notlessdblequal put dup 21 /notgreaterdblequal put dup 22 /precedenotslnteql put dup 23 /follownotslnteql put dup 24 /precedenotdbleqv put dup 25 /follownotdbleqv put dup 26 /lessnotdblequal put dup 27 /greaternotdblequal put dup 28 /notsimilar put dup 29 /notapproxequal put dup 30 /upslope put dup 31 /downslope put dup 32 /notsubsetoreql put dup 33 /notsupersetoreql put dup 34 /notsubsetordbleql put dup 35 /notsupersetordbleql put dup 36 /subsetornotdbleql put dup 37 /supersetornotdbleql put dup 38 /subsetornoteql put dup 39 /supersetornoteql put dup 40 /subsetnoteql put dup 41 /supersetnoteql put dup 42 /notsubseteql put dup 43 /notsuperseteql put dup 44 /notparallel put dup 45 /notbar put dup 46 /notshortbar put dup 47 /notshortparallel put dup 48 /notturnstile put dup 49 /notforces put dup 50 /notsatisfies put dup 51 /notforcesextra put dup 52 /nottriangeqlright put dup 53 /nottriangeqlleft put dup 54 /nottriangleleft put dup 55 /nottriangleright put dup 56 /notarrowleft put dup 57 /notarrowright put dup 58 /notdblarrowleft put dup 59 /notdblarrowright put dup 60 /notdblarrowboth put dup 61 /notarrowboth put dup 62 /dividemultiply put dup 63 /emptyset put dup 64 /notexistential put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /hatwide put dup 92 /hatwider put dup 93 /tildewide put dup 94 /tildewider put dup 96 /Finv put dup 97 /Gmir put dup 102 /Omegainv put dup 103 /eth put dup 104 /equalorsimilar put dup 105 /beth put dup 106 /gimel put dup 107 /daleth put dup 108 /lessdot put dup 109 /greaterdot put dup 110 /multicloseleft put dup 111 /multicloseright put dup 112 /barshort put dup 113 /parallelshort put dup 114 /integerdivide put dup 115 /similar put dup 116 /approxequal put dup 117 /approxorequal put dup 118 /followsorequal put dup 119 /precedesorequal put dup 120 /archleftdown put dup 121 /archrightdown put dup 122 /Digamma put dup 123 /kappa put dup 124 /k put dup 125 /planckover2pi put dup 126 /planckover2pi1 put dup 127 /epsiloninv put dup 128 /notsubsetoreql put dup 160 /space put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT10 %!PS-AdobeFont-1.1: CMTT10 1.00B %%CreationDate: 1992 Apr 26 10:42:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /visiblespace put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /arrowup put dup 12 /arrowdown put dup 13 /quotesingle put dup 14 /exclamdown put dup 15 /questiondown put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /visiblespace put dup 33 /exclam put dup 34 /quotedbl put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /less put dup 61 /equal put dup 62 /greater put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /backslash put dup 93 /bracketright put dup 94 /asciicircum put dup 95 /underscore put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /braceleft put dup 124 /bar put dup 125 /braceright put dup 126 /asciitilde put dup 127 /dieresis put dup 128 /visiblespace put dup 160 /space put readonly def /FontBBox{-4 -235 731 800}readonly def /UniqueID 5000832 def currentdict end currentfile eexec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Fl(-algebra)g(coincides)g(with)f(the)g(\014rst)h (cohomol-)150 1659 y(ogy)42 b(of)g(the)h(susp)s(ension)h(space)g ([HPS].)d(Giordano,)j(Putnam)f(and)g(Sk)-6 b(au)43 b([GPS])f(used)i (the)150 1778 y(\(ordered\))32 b Fh(K)7 b Fl(-groups)32 b(of)f(the)h(C)1384 1742 y Fe(\003)1430 1778 y Fl(-algebra)f(to)g(c)m (haracterize)h(orbit)f(equiv)-6 b(alence)32 b(for)g(minimal)150 1898 y Fg(Z)p Fl(-subshifts.)349 2017 y(In)k(a)f(similar)g(v)m(ein)g (Connes)h([Con])f(used)i(P)m(enrose)f(tilings)f(to)g(motiv)-6 b(ate)34 b(ideas)i(on)f(non-)150 2137 y(comm)m(utativ)m(e)k(top)s (ology)-8 b(.)62 b(Sp)s(eci\014cally)-8 b(,)40 b(he)h(considered)g(the) f(quotien)m(t)f Fh(X)47 b Fl(of)40 b(the)g(space)h(of)150 2257 y(P)m(enrose)j(tilings)f(b)m(y)g(the)g(equiv)-6 b(alence)44 b(relation)f(of)g(translation.)73 b(Using)43 b(the)h(hierarc)m(hical)150 2376 y(structure)35 b(of)g(the)g(P)m (enrose)g(tilings)f(there)h(is)f(a)h(natural)f(homeomorphism)h(b)s(et)m (w)m(een)h Fh(X)41 b Fl(and)150 2496 y(a)35 b(quotien)m(t)g Fh(Y)57 b Fl(of)35 b(sequences)j(mo)s(dulo)d(another)g(equiv)-6 b(alence)36 b(relation.)49 b(As)35 b(is)g(common)h(for)150 2615 y(equiv)-6 b(alence)37 b(relations)e(coming)h(from)g(dynamics,)h (the)f(quotien)m(t)g(top)s(ology)e(of)i Fh(Y)58 b Fl(is)36 b(trivial.)150 2735 y(And)e(so)f(Connes)i(considers)g(a)e(\(noncomm)m (utativ)m(e\))g(C)2257 2699 y Fe(\003)2303 2735 y Fl(-algebra)g(asso)s (ciated)h(with)f(the)h(quo-)150 2854 y(tien)m(t)e Fh(Y)22 b Fl(.)43 b(The)33 b Fh(K)7 b Fl(-theory)31 b(of)h(this)g(C)1552 2818 y Fe(\003)1598 2854 y Fl(-algebra)g(is)g(his)g(substitute)g(for)h (the)f(\(trivial\))d(quotien)m(t)150 2974 y(top)s(ology)-8 b(.)349 3093 y(This)45 b(w)m(as)f(pushed)j(one)d(step)h(further)h(b)m (y)e(Kellendonk)h(and)g(Putnam)f([Ke1-4,)j(KeP)-8 b(,)150 3213 y(AnP])28 b(in)g(whic)m(h)h(examples)g(of)f(tiling)g(dynamical)f (systems)i(suc)m(h)h(as)e(the)h(P)m(enrose)g(system)g(are)150 3333 y(considered)44 b(and)g(their)f(\(unordered\))h Fh(K)7 b Fl(-theory)42 b(is)h(w)m(ork)m(ed)g(out,)i(obtaining)d(for)h (instance)150 3452 y(cohomology)32 b(groups)i(for)g(the)f(spaces)i(of)e (tilings.)349 3572 y(W)-8 b(e)27 b(consider)g(substitution)g(tiling)e (systems)i(in)g Fg(R)2241 3535 y Fj(d)2287 3572 y Fl(,)g(suc)m(h)h(as)f (the)f(P)m(enrose)i(tilings)d(in)i Fg(R)3678 3535 y Fk(2)3722 3572 y Fl(,)150 3691 y(as)35 b(dynamical)g(systems)h(with)e Fg(R)1427 3655 y Fj(d)1509 3691 y Fl(\(translation\))g(actions.)49 b(These)37 b(are)e(natural)g(generaliza-)150 3811 y(tions)i(of)g(the)h (susp)s(ensions)h(of)f Fg(Z)1415 3775 y Fj(d)1499 3811 y Fl(\(substitution\))f(subshifts.)58 b(\(W)-8 b(e)37 b(will)f(restrict)h(atten)m(tion)150 3930 y(to)42 b Fh(d)i Fl(=)g(2;)j(while)42 b(most)h(of)g(our)g(metho)s(ds)g(carry)f(o)m(v)m (er)h(to)f(higher)i(dimensions,)i(there)d(are)150 4050 y(complications)30 b(whic)m(h)g(w)m(e)h(wish)f(to)g(a)m(v)m(oid.\))42 b(As)30 b(noted)g(ab)s(o)m(v)m(e,)h(susp)s(ensions)h(are)e(sometimes) 150 4169 y(in)m(tro)s(duced)36 b(to)d(get)i(in)m(v)-6 b(arian)m(ts)34 b(to)g(distinguish)h(b)s(et)m(w)m(een)h(subshifts,)g (insofar)f(as)f(the)h(spaces)150 4289 y(on)24 b(whic)m(h)h(the)g (subshifts)h(act)e(are)g(all)f(the)h(same)h(|)f(Can)m(tor)g(sets.)41 b(F)-8 b(or)24 b(tilings)f(the)i(natural)f(ob-)150 4408 y(jects)29 b(already)e(ha)m(v)m(e)i Fg(R)1018 4372 y Fj(d)1092 4408 y Fl(actions)f(and)g(di\013eren)m(t)h(tiling)e(spaces)i (need)g(not)f(b)s(e)h(homeomorphic.)150 4528 y(In)42 b(particular,)h(the)f(cohomology)f(groups)h(of)g(suc)m(h)h(spaces)g (are)f(in)m(v)-6 b(arian)m(ts)42 b(of)g(top)s(ological)150 4648 y(conjugacy)-8 b(.)46 b(F)-8 b(or)33 b(1-dimensional)h(tiling)f (systems,)h(Barge)f(and)h(Diamond)g([BD])f(w)m(ork)m(ed)h(out)150 4767 y(complete)g(homeomorphism)g(in)m(v)-6 b(arian)m(ts.)349 4887 y(Although)30 b(in)g(principle)g(it)g(should)h(b)s(e)f(p)s (ossible)g(to)g(w)m(ork)f(out)h(the)g(cohomology)f(groups)150 5006 y(for)k(tiling)f(spaces,)j(in)e(practice)g(it)g(can)h(b)s(e)f (di\016cult)g(since)h(the)g(n)m(um)m(b)s(er)h(of)e(di\013eren)m(t)h (tiles)f(in)150 5126 y(in)m(teresting)e(mo)s(dels)g(\(esp)s(ecially)f (after)h(collaring)f(|)h(see)g(b)s(elo)m(w\),)g(whic)m(h)h(giv)m(es)e (the)h(n)m(um)m(b)s(er)150 5245 y(of)k(generators)g(for)f(the)h(c)m (hain)g(groups,)g(is)g(usually)f(large,)g(ev)m(en)h(in\014nite)g(for)g (examples)f(suc)m(h)1925 5519 y(1)p eop %%Page: 2 3 2 2 bop 150 100 a Fl(as)35 b(pin)m(wheel)h(tilings.)49 b(New)35 b(in)g(this)g(w)m(ork)f(is)h(analysis)g(of)g(t)m(w)m(o)g(asp)s (ects)g(of)h(the)f(cohomology)150 219 y(of)h(substitution)g(tiling)f (spaces)i(that)e(are)h(b)s(oth)f(easy)h(to)f(compute)i(and)f (informativ)m(e)f(ab)s(out)150 339 y(the)f(top)s(ology)d(of)j(the)f (spaces.)349 458 y(The)28 b(\014rst)g(asp)s(ect)g(is)f(an)g(order)h (structure)g(for)g(the)g(top)f(dimensional)2989 433 y(\024)2978 458 y(Cec)m(h)i(cohomology)-8 b(.)150 578 y(F)g(or)39 b(some)g(history)g(in)g(a)h(\\noncomm)m(utativ)m(e")f(setting)g(see)h 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y(the)j(area)f(stretc)m(hing)h(factor)g(of)f(the)h(substitution.) 51 b(W)-8 b(e)35 b(sho)m(w)h(that)f(the)h(p)s(ositiv)m(e)f(cone)h(is)f (a)150 1534 y(homeomorphism)d(in)m(v)-6 b(arian)m(t,)31 b(and)g(that)g(one)h(can)f(therefore)h(extract)f(in)m(v)-6 b(arian)m(t)31 b(top)s(ological)150 1654 y(information)i(from)g(the)h (easily)f(computable)g(stretc)m(hing)h(factor)g Fh(\025)p Fl(.)349 1773 y(W)-8 b(e)51 b(pro)m(v)m(e)g(results)f(of)h(this)f(form) h(for)f(t)m(w)m(o)g(classes)i(of)e(tiling)f(spaces.)97 b(F)-8 b(or)49 b(\\\014xed-)150 1893 y(orien)m(tation")36 b(tiling)f(spaces)j(\(denoted)f Fh(X)1783 1908 y Fj(x)1869 1893 y Fl(b)s(elo)m(w\),)g(w)m(e)g(need)h(the)f(additional)e (assumption)150 2012 y(that)e(tiles)g(only)g(app)s(ear)h(in)f(a)h (\014nite)f(n)m(um)m(b)s(er)i(of)f(distinct)g(orien)m(tations)f(in)g (eac)m(h)i(tiling.)43 b(F)-8 b(or)150 2132 y(\\all-orien)m(tation")35 b(tiling)g(spaces)j(\(denoted)f Fh(X)1971 2147 y Fj(\036)2060 2132 y Fl(b)s(elo)m(w\),)f(this)h(assumption)g(is)f(not)g(needed.)150 2252 y(Our)e(main)f(result)h(is)150 2431 y Fm(Theorem)40 b(1.)51 b Fd(L)-5 b(et)38 b Fh(X)44 b Fd(and)38 b Fh(Y)59 b Fd(b)-5 b(e)38 b(substitution)f(tiling)g(systems)g(whose)h (substitutions)f(have)150 2551 y(ar)-5 b(e)g(a)50 b(str)-5 b(etching)49 b(factors)h Fh(\025)g Fd(and)h Fh(\025)1569 2515 y Fe(0)1597 2551 y Fd(,)j(r)-5 b(esp)g(e)g(ctively.)89 b(Supp)-5 b(ose)50 b(that)g(either)g(i\))g Fh(X)58 b Fd(and)50 b Fh(Y)150 2670 y Fd(have)41 b(\014nite)f(r)-5 b(elative)41 b(orientation)f(gr)-5 b(oups)41 b(and)g(the)h(\014xe)-5 b(d-orientation)39 b(sp)-5 b(ac)g(es)40 b Fh(X)3354 2685 y Fj(x)3445 2670 y Fd(and)h Fh(Y)3702 2685 y Fj(y)150 2790 y Fd(ar)-5 b(e)36 b(home)-5 b(omorphic,)36 b(or)g(ii\))f(the)i(al) 5 b(l-orientation)35 b(sp)-5 b(ac)g(es)36 b Fh(X)2492 2805 y Fj(\036)2580 2790 y Fd(and)h Fh(Y)2833 2805 y Fj(\036)2922 2790 y Fd(ar)-5 b(e)36 b(home)-5 b(omorphic.)150 2909 y(Then)166 3029 y(\(1\))50 b(If)34 b Fh(\025)h 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661 y Fe(\003)1440 697 y Fl(\()p Fh(X)1562 712 y Fj(x)1611 697 y Fl(\))g(the)g(structure)i (of)e(a)g(mo)s(dule)g(o)m(v)m(er)h(the)f(group)g(ring,)150 817 y(and)f(this)g(mo)s(dule)g(structure)h(is)f(quite)g(useful.)59 b(In)38 b(particular,)g(for)g(an)m(y)g(realization)f(of)h Fh(X)3700 832 y Fj(x)150 936 y Fl(as)g(the)h(in)m(v)m(erse)g(limit)e (of)h(simplicial)g(complexes,)h(as)g(in)f([AnP],)f(the)i(\(co\)c)m (hain)f(groups)h(are)150 1056 y(themselv)m(es)28 b(mo)s(dules,)g(and)g (the)f(b)s(oundary)g(and)h(substitution)f(maps)g(are)g(equiv)-6 b(arian)m(t.)42 b(This)150 1176 y(greatly)c(simpli\014es)h(the)h (calculations.)60 b(F)-8 b(or)39 b(P)m(enrose)h(tiles)e(for)i(example)f ([AnP],)e(there)j(are)150 1295 y(4)46 b(t)m(yp)s(es)h(of)f(tiles,)j (eac)m(h)f(app)s(earing)e(in)h(10)f(di\013eren)m(t)h(orien)m(tations.) 83 b(Without)47 b(using)g(the)150 1415 y(rotational)c(symmetry)h(the)h (action)g(of)g(substitution)g(on)g(2-c)m(hains)h(w)m(ould)f(b)s(e)g (giv)m(en)f(b)m(y)h(a)150 1534 y(40)27 b Ff(\002)i Fl(40)41 b(matrix.)68 b(Using)42 b(the)g(rotational)e(symmetry)i(w)m(e)g (express)h(it)e(instead)h(as)g(a)f(4)28 b Ff(\002)g Fl(4)150 1654 y(matrix)i(with)h(en)m(tries)h(in)f(the)g(group)h(ring)f(of)g Fg(Z)1958 1669 y Fk(10)2043 1654 y Fl(,)g(whic)m(h)h(can)g(b)s(e)f (analyzed)g(one)h(irreducible)150 1773 y(represen)m(tation)37 b(at)e(a)h(time.)52 b(The)36 b(substitutions)h(on)f(0-)g(and)g(1-c)m (hains,)h(and)f(the)h(b)s(oundary)150 1893 y(maps,)49 b(are)c(handled)i(similarly)-8 b(.)78 b(The)46 b(result)g(is)f(a)g (streamlined)h(calculation)f(that)g(yields)150 2012 y(information)31 b(b)s(ey)m(ond)h(that)g(obtained)f(b)m(y)h(older)g(metho)s(ds.)44 b(Anderson)32 b(and)g(Putnam)g([AnP])150 2132 y(computed)23 b Fh(H)689 2096 y Fk(0)734 2132 y Fl(\()p Fh(X)856 2147 y Fj(x)905 2132 y Fl(\))28 b(=)g Fg(Z)p Fl(,)c Fh(H)1286 2096 y Fk(1)1331 2132 y Fl(\()p Fh(X)1453 2147 y Fj(x)1502 2132 y Fl(\))k(=)g Fg(Z)1740 2096 y Fk(5)1807 2132 y Fl(and)23 b Fh(H)2081 2096 y Fk(2)2126 2132 y Fl(\()p Fh(X)2248 2147 y Fj(x)2297 2132 y Fl(\))k(=)i Fg(Z)2535 2096 y Fk(8)2580 2132 y Fl(.)40 b(W)-8 b(e)22 b(note)h(that,)h(as)e(mo) s(dules,)150 2252 y(a)g(\014nite-index)i(additiv)m(e)e(subgroup)i(of)f Fh(H)1722 2215 y Fk(1)1766 2252 y Fl(\()p Fh(X)1888 2267 y Fj(x)1938 2252 y Fl(\))f(equals)h Fg(Z)p Fl([)p Fh(t)p Fl(])p Fh(=)p Fl(\()p Fh(t)q Ff(\000)q Fl(1\))q Ff(\010)q Fg(Z)p Fl([)p Fh(t)p Fl(])p Fh(=)p Fl(\()p Fh(t)3104 2215 y Fk(4)3148 2252 y Ff(\000)q Fh(t)3262 2215 y Fk(3)3308 2252 y Fl(+)q Fh(t)3422 2215 y Fk(2)3468 2252 y Ff(\000)q Fh(t)q Fl(+)q(1\))150 2371 y(while)38 b(a)f(\014nite)h(index)g (subgroup)h(of)f Fh(H)1680 2335 y Fk(2)1724 2371 y Fl(\()p Fh(X)1846 2386 y Fj(x)1896 2371 y Fl(\))f(equals)h(\()p Fg(Z)p Fl([)p Fh(t)p Fl(])p Fh(=)p Fl(\()p Fh(t)24 b Ff(\000)h Fl(1\)\))2855 2335 y Fk(2)2924 2371 y Ff(\010)h Fl(\()p Fg(Z)p Fl([)p Fh(t)p Fl(])p Fh(=)p Fl(\()p Fh(t)e Fl(+)h(1\)\))3603 2335 y Fk(2)3673 2371 y Ff(\010)150 2491 y Fg(Z)p Fl([)p Fh(t)p Fl(])p Fh(=)p Fl(\()p Fh(t)433 2455 y Fk(4)499 2491 y Ff(\000)e Fh(t)635 2455 y Fk(3)702 2491 y Fl(+)f Fh(t)837 2455 y Fk(2)904 2491 y Ff(\000)h Fh(t)f Fl(+)g(1\),)33 b(where)h(rotation)e(b)m(y)h(2)p Fh(\031)t(=)p Fl(10)f(is)h(m)m(ultiplication)g(b)m(y)g Fh(t)p Fl(.)150 2730 y Fm(2.)51 b(Tiling)37 b(spaces)h(and)f(their)g (asso)s(ciated)g(in)m(v)m(erse)g(limits.)349 2850 y Fl(Before)c (de\014ning)h(substitution)f(tiling)e(spaces)j(in)f(general)g(w)m(e)g (presen)m(t)h(some)f(examples)150 2969 y(whose)h(top)s(ological)d(prop) s(erties)j(w)m(e)g(will)e(later)h(consider.)349 3089 y(A)43 b(\\c)m(hair")h(tiling)e(of)i(the)f(plane,)j(Fig.)d(1,)i(can)f (b)s(e)g(made)g(as)f(follo)m(ws.)75 b(Consider)44 b(the)150 3208 y(L-shap)s(ed)g(tile)d(of)i(Fig.)e(2.)71 b(Divide)42 b(this)g(tile)g(\(also)g(called)g(a)g(\\tile)f(of)i(lev)m(el)f(0"\))f (in)m(to)h(four)150 3328 y(pieces)34 b(as)f(in)g(Fig.)e(3)i(and)g (rescale)h(b)m(y)f(a)f(linear)h(factor)g(of)g(2)f(so)h(that)f(eac)m(h)i (piece)g(is)e(the)i(same)150 3448 y(size)40 b(as)g(the)g(original.)62 b(This)40 b(yields)f(a)g(collection)g(of)h(4)g(tiles)f(that)g(w)m(e)h (call)f(a)h(\\tile)e(of)i(lev)m(el)150 3567 y(1".)54 b(Sub)s(dividing)37 b(eac)m(h)h(of)f(these)h(tiles)e(and)i(rescaling)f (giv)m(es)f(a)h(collection)f(of)h(16)g(tiles)f(that)150 3687 y(w)m(e)e(call)f(a)g(tile)g(of)g(lev)m(el)g(2.)44 b(Rep)s(eating)34 b(the)g(pro)s(cess)g Fh(n)g Fl(times)f(giv)m(es)g(a)g (collection)g(of)h(4)3485 3650 y Fj(n)3572 3687 y Fl(tiles)150 3806 y(|)h(a)g(tile)g(of)h(lev)m(el)f Fh(n)p Fl(.)50 b(A)35 b(\\c)m(hair")g(tiling)g(is)g(a)g(tiling)g(of)g(the)h(plane)g (with)f(the)h(prop)s(ert)m(y)f(that)150 3926 y(ev)m(ery)42 b(\014nite)h(sub)s(collection)g(of)f(tiles)g(is)h(congruen)m(t)g(to)f (a)g(subset)i(of)e(a)h(tile)e(of)i(some)g(lev)m(el.)150 4045 y(A)36 b(c)m(hair)g(tiling)f(has)i(only)e(one)h(t)m(yp)s(e)g(of)h (tile,)e(app)s(earing)i(in)f(4)f(di\013eren)m(t)j(orien)m(tations.)52 b(\(F)-8 b(or)150 4165 y(c)m(hair)34 b(tilings,)f(and)i(in)e(fact)i(v)m (ery)e(generally)g(for)i(tilings)e(made)h(b)m(y)g(suc)m(h)h (substitution)g(rules,)150 4284 y(there)g(are)f(\\matc)m(hing)f(rules") i(whic)m(h)f(pro)m(vide)h(a)e(di\013eren)m(t)i(metho)s(d)f(of)h (construction)f([Go)s(o,)150 4404 y(Ra2].\))349 4523 y(Somewhat)43 b(more)f(complicated)g(are)h(the)f(\\P)m(enrose")h (tilings,)g(Fig.)f(4,)i(whic)m(h)f(ha)m(v)m(e)f(4)150 4643 y(di\013eren)m(t)30 b(t)m(yp)s(es)g(of)g(triangular)e(tiles,)i (eac)m(h)g(app)s(earing)g(in)f(10)g(di\013eren)m(t)i(orien)m(tations.) 42 b(The)30 b(4)150 4763 y(tiles,)d(and)g(the)f(rule)h(for)f(sub)s (dividing)g(them,)i(are)e(sho)m(wn)h(in)g(Fig.)e(5.)41 b(\(These)27 b(tiles)f(are)h(not)f(the)150 4882 y(familiar)33 b(kites)g(and)i(darts.)46 b(Ho)m(w)m(ev)m(er,)34 b(the)g(space)h(of)f (substitution)h(tilings)e(dev)m(elop)s(ed)i(from)150 5002 y(these)42 b(4)f(triangles)g(is)g(homeomorphic)h(to)f(the)h(space) g(of)g(P)m(enrose)g(kite)e(and)i(dart)f(tilings\).)150 5121 y(Here)34 b(the)f(linear)g(stretc)m(hing)h(factor)f(is)h(the)f (golden)h(mean)f Fh(\034)39 b Fl(=)28 b(\(1)22 b(+)2833 5039 y Ff(p)p 2916 5039 50 4 v 82 x Fl(5\))p Fh(=)p Fl(2,)32 b(and)i(the)g(area)1925 5519 y(3)p eop %%Page: 4 5 4 4 bop -193 4215 a @beginspecial -9 @llx 148 @lly 622 @urx 644 @ury 4320 @rwi @setspecial %%BeginDocument: fig1.ps 50 dict begin % This is a standard prolog for Postscript generated by Tk's canvas % widget. % @(#) prolog.ps 1.2 94/12/09 10:53:18 % The definitions below just define all of the variables used in % any of the procedures here. This is needed for obscure reasons % explained on p. 716 of the Postscript manual (Section H.2.7, % "Initializing Variables," in the section on Encapsulated Postscript). /baseline 0 def /stipimage 0 def /height 0 def /justify 0 def /lineLength 0 def /spacing 0 def /stipple 0 def /strings 0 def /xoffset 0 def /yoffset 0 def /tmpstip null def % Define the array ISOLatin1Encoding (which specifies how characters are % encoded for ISO-8859-1 fonts), if it isn't already present (Postscript % level 2 is supposed to define it, but level 1 doesn't). systemdict /ISOLatin1Encoding known not { /ISOLatin1Encoding [ /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /dotlessi /grave /acute /circumflex /tilde /macron /breve /dotaccent /dieresis /space /ring /cedilla /space /hungarumlaut /ogonek /caron /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedillar /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def } if % Override the setfont procedure with a new procedure that re-encodes % the font to use the ISO Latin-1 style. The body of this procedure % comes from Section 5.6.1 of the Postscript book. /realsetfont /setfont load def /setfont { dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end % I'm not sure why it's necessary to use "definefont" on this new % font, but it seems to be important; just use the name "Temporary" % for the font. /Temporary exch definefont realsetfont } bind def % StrokeClip % % This procedure converts the current path into a clip area under % the assumption of stroking. It's a bit tricky because some Postscript % interpreters get errors during strokepath for dashed lines. If % this happens then turn off dashes and try again. /StrokeClip { {strokepath} stopped { (This Postscript printer gets limitcheck overflows when) = (stippling dashed lines; lines will be printed solid instead.) = [] 0 setdash strokepath} if clip } bind def % desiredSize EvenPixels closestSize % % The procedure below is used for stippling. Given the optimal size % of a dot in a stipple pattern in the current user coordinate system, % compute the closest size that is an exact multiple of the device's % pixel size. This allows stipple patterns to be displayed without % aliasing effects. /EvenPixels { % Compute exact number of device pixels per stipple dot. dup 0 matrix currentmatrix dtransform dup mul exch dup mul add sqrt % Round to an integer, make sure the number is at least 1, and compute % user coord distance corresponding to this. dup round dup 1 lt {pop 1} if exch div mul } bind def % width height string StippleFill -- % % Given a path already set up and a clipping region generated from % it, this procedure will fill the clipping region with a stipple % pattern. "String" contains a proper image description of the % stipple pattern and "width" and "height" give its dimensions. Each % stipple dot is assumed to be about one unit across in the current % user coordinate system. This procedure trashes the graphics state. /StippleFill { % The following code is needed to work around a NeWSprint bug. /tmpstip 1 index def % Change the scaling so that one user unit in user coordinates % corresponds to the size of one stipple dot. 1 EvenPixels dup scale % Compute the bounding box occupied by the path (which is now % the clipping region), and round the lower coordinates down % to the nearest starting point for the stipple pattern. Be % careful about negative numbers, since the rounding works % differently on them. pathbbox 4 2 roll 5 index div dup 0 lt {1 sub} if cvi 5 index mul 4 1 roll 6 index div dup 0 lt {1 sub} if cvi 6 index mul 3 2 roll % Stack now: width height string y1 y2 x1 x2 % Below is a doubly-nested for loop to iterate across this area % in units of the stipple pattern size, going up columns then % across rows, blasting out a stipple-pattern-sized rectangle at % each position 6 index exch { 2 index 5 index 3 index { % Stack now: width height string y1 y2 x y gsave 1 index exch translate 5 index 5 index true matrix tmpstip imagemask grestore } for pop } for pop pop pop pop pop } bind def % -- AdjustColor -- % Given a color value already set for output by the caller, adjusts % that value to a grayscale or mono value if requested by the CL % variable. /AdjustColor { CL 2 lt { currentgray CL 0 eq { .5 lt {0} {1} ifelse } if setgray } if } bind def % x y strings lineLength spacing xoffset yoffset justify stipple DrawText -- % This procedure does all of the real work of drawing text. The % color and font must already have been set by the caller, and the % following arguments must be on the stack: % % x, y - Coordinates at which to draw text. % strings - An array of strings, one for each line of the text item, % in order from top to bottom. % lineLength - Minimum line length: needed to justify text properly. % spacing - Spacing between lines. % xoffset - Horizontal offset for text bbox relative to x and y: 0 for % nw/w/sw anchor, -0.5 for n/center/s, and -1.0 for ne/e/se. % yoffset - Vertical offset for text bbox relative to x and y: 0 for % nw/n/ne anchor, +0.5 for w/center/e, and +1.0 for sw/s/se. % justify - 0 for left justification, 0.5 for center, 1 for right justify. % stipple - Boolean value indicating whether or not text is to be % drawn in stippled fashion. If text is stippled, % procedure StippleText must have been defined to call % StippleFill in the right way. % % Also, when this procedure is invoked, the color and font must already % have been set for the text. /DrawText { /stipple exch def /justify exch def /yoffset exch def /xoffset exch def /spacing exch def /lineLength exch def /strings exch def % First scan through all of the text to find the widest line (if it's % longer than the "lineLength" argument). strings { stringwidth pop dup lineLength gt {/lineLength exch def} {pop} ifelse newpath } forall % Compute the baseline offset and the actual font height. 0 0 moveto (TXygqPZ) false charpath pathbbox dup /baseline exch def exch pop exch sub /height exch def pop newpath % Translate coordinates first so that the origin is at the upper-left % corner of the text's bounding box. Remember that x and y for % positioning are still on the stack. translate lineLength xoffset mul strings length 1 sub spacing mul height add yoffset mul translate % Now use the baseline and justification information to translate so % that the origin is at the baseline and positioning point for the % first line of text. justify lineLength mul baseline neg translate % Iterate over each of the lines to output it. 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%%Page: 5 6 5 5 bop 1410 1299 a @beginspecial 203 @llx 293 @lly 409 @urx 499 @ury 1080 @rwi @setspecial %%BeginDocument: fig2.ps 50 dict begin % This is a standard prolog for Postscript generated by Tk's canvas % widget. % SCCS: @(#) prolog.ps 1.5 96/02/17 17:45:11 % The definitions below just define all of the variables used in % any of the procedures here. This is needed for obscure reasons % explained on p. 716 of the Postscript manual (Section H.2.7, % "Initializing Variables," in the section on Encapsulated Postscript). /baseline 0 def /stipimage 0 def /height 0 def /justify 0 def /lineLength 0 def /spacing 0 def /stipple 0 def /strings 0 def /xoffset 0 def /yoffset 0 def /tmpstip null def % Define the array ISOLatin1Encoding (which specifies how characters are % encoded for ISO-8859-1 fonts), if it isn't already present (Postscript % level 2 is supposed to define it, but level 1 doesn't). systemdict /ISOLatin1Encoding known not { /ISOLatin1Encoding [ /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /dotlessi /grave /acute /circumflex /tilde /macron /breve /dotaccent /dieresis /space /ring /cedilla /space /hungarumlaut /ogonek /caron /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedillar /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def } if % font ISOEncode font % This procedure changes the encoding of a font from the default % Postscript encoding to ISOLatin1. It's typically invoked just % before invoking "setfont". The body of this procedure comes from % Section 5.6.1 of the Postscript book. /ISOEncode { dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end % I'm not sure why it's necessary to use "definefont" on this new % font, but it seems to be important; just use the name "Temporary" % for the font. /Temporary exch definefont } bind def % StrokeClip % % This procedure converts the current path into a clip area under % the assumption of stroking. It's a bit tricky because some Postscript % interpreters get errors during strokepath for dashed lines. If % this happens then turn off dashes and try again. /StrokeClip { {strokepath} stopped { (This Postscript printer gets limitcheck overflows when) = (stippling dashed lines; lines will be printed solid instead.) = [] 0 setdash strokepath} if clip } bind def % desiredSize EvenPixels closestSize % % The procedure below is used for stippling. Given the optimal size % of a dot in a stipple pattern in the current user coordinate system, % compute the closest size that is an exact multiple of the device's % pixel size. This allows stipple patterns to be displayed without % aliasing effects. /EvenPixels { % Compute exact number of device pixels per stipple dot. dup 0 matrix currentmatrix dtransform dup mul exch dup mul add sqrt % Round to an integer, make sure the number is at least 1, and compute % user coord distance corresponding to this. dup round dup 1 lt {pop 1} if exch div mul } bind def % width height string StippleFill -- % % Given a path already set up and a clipping region generated from % it, this procedure will fill the clipping region with a stipple % pattern. "String" contains a proper image description of the % stipple pattern and "width" and "height" give its dimensions. Each % stipple dot is assumed to be about one unit across in the current % user coordinate system. This procedure trashes the graphics state. /StippleFill { % The following code is needed to work around a NeWSprint bug. /tmpstip 1 index def % Change the scaling so that one user unit in user coordinates % corresponds to the size of one stipple dot. 1 EvenPixels dup scale % Compute the bounding box occupied by the path (which is now % the clipping region), and round the lower coordinates down % to the nearest starting point for the stipple pattern. Be % careful about negative numbers, since the rounding works % differently on them. pathbbox 4 2 roll 5 index div dup 0 lt {1 sub} if cvi 5 index mul 4 1 roll 6 index div dup 0 lt {1 sub} if cvi 6 index mul 3 2 roll % Stack now: width height string y1 y2 x1 x2 % Below is a doubly-nested for loop to iterate across this area % in units of the stipple pattern size, going up columns then % across rows, blasting out a stipple-pattern-sized rectangle at % each position 6 index exch { 2 index 5 index 3 index { % Stack now: width height string y1 y2 x y gsave 1 index exch translate 5 index 5 index true matrix tmpstip imagemask grestore } for pop } for pop pop pop pop pop } bind def % -- AdjustColor -- % Given a color value already set for output by the caller, adjusts % that value to a grayscale or mono value if requested by the CL % variable. /AdjustColor { CL 2 lt { currentgray CL 0 eq { .5 lt {0} {1} ifelse } if setgray } if } bind def % x y strings spacing xoffset yoffset justify stipple DrawText -- % This procedure does all of the real work of drawing text. The % color and font must already have been set by the caller, and the % following arguments must be on the stack: % % x, y - Coordinates at which to draw text. % strings - An array of strings, one for each line of the text item, % in order from top to bottom. % spacing - Spacing between lines. % xoffset - Horizontal offset for text bbox relative to x and y: 0 for % nw/w/sw anchor, -0.5 for n/center/s, and -1.0 for ne/e/se. % yoffset - Vertical offset for text bbox relative to x and y: 0 for % nw/n/ne anchor, +0.5 for w/center/e, and +1.0 for sw/s/se. % justify - 0 for left justification, 0.5 for center, 1 for right justify. % stipple - Boolean value indicating whether or not text is to be % drawn in stippled fashion. If text is stippled, % procedure StippleText must have been defined to call % StippleFill in the right way. % % Also, when this procedure is invoked, the color and font must already % have been set for the text. /DrawText { /stipple exch def /justify exch def /yoffset exch def /xoffset exch def /spacing exch def /strings exch def % First scan through all of the text to find the widest line. /lineLength 0 def strings { stringwidth pop dup lineLength gt {/lineLength exch def} {pop} ifelse newpath } forall % Compute the baseline offset and the actual font height. 0 0 moveto (TXygqPZ) false charpath pathbbox dup /baseline exch def exch pop exch sub /height exch def pop newpath % Translate coordinates first so that the origin is at the upper-left % corner of the text's bounding box. Remember that x and y for % positioning are still on the stack. translate lineLength xoffset mul strings length 1 sub spacing mul height add yoffset mul translate % Now use the baseline and justification information to translate so % that the origin is at the baseline and positioning point for the % first line of text. justify lineLength mul baseline neg translate % Iterate over each of the lines to output it. For each line, % compute its width again so it can be properly justified, then % display it. strings { dup stringwidth pop justify neg mul 0 moveto stipple { % The text is stippled, so turn it into a path and print % by calling StippledText, which in turn calls StippleFill. % Unfortunately, many Postscript interpreters will get % overflow errors if we try to do the whole string at % once, so do it a character at a time. gsave /char (X) def { char 0 3 -1 roll put currentpoint gsave char true charpath clip StippleText grestore char stringwidth translate moveto } forall grestore } {show} ifelse 0 spacing neg translate } forall } bind def /CL 1 def save 306.0 396.0 translate 0.9589 0.9589 scale -107 -107 translate 0 214 moveto 214 214 lineto 214 0 lineto 0 0 lineto closepath clip newpath gsave 100 114 moveto 100 24 lineto 10 24 lineto 10 204 lineto 190 204 lineto 190 114 lineto 100 114 lineto 1 setlinewidth 1 setlinecap 1 setlinejoin 0.000 0.000 0.000 setrgbcolor AdjustColor stroke grestore restore showpage end %%EndDocument @endspecial 1437 1718 a Fl(Figure)33 b(2.)44 b(The)34 b(c)m(hair)g(tile)1266 5052 y @beginspecial 68 @llx 217 @lly 544 @urx 576 @ury 3600 @rwi @setspecial %%BeginDocument: fig3.ps 50 dict begin % This is a standard prolog for Postscript generated by Tk's canvas % widget. % SCCS: @(#) prolog.ps 1.5 96/02/17 17:45:11 % The definitions below just define all of the variables used in % any of the procedures here. This is needed for obscure reasons % explained on p. 716 of the Postscript manual (Section H.2.7, % "Initializing Variables," in the section on Encapsulated Postscript). /baseline 0 def /stipimage 0 def /height 0 def /justify 0 def /lineLength 0 def /spacing 0 def /stipple 0 def /strings 0 def /xoffset 0 def /yoffset 0 def /tmpstip null def % Define the array ISOLatin1Encoding (which specifies how characters are % encoded for ISO-8859-1 fonts), if it isn't already present (Postscript % level 2 is supposed to define it, but level 1 doesn't). systemdict /ISOLatin1Encoding known not { /ISOLatin1Encoding [ /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /dotlessi /grave /acute /circumflex /tilde /macron /breve /dotaccent /dieresis /space /ring /cedilla /space /hungarumlaut /ogonek /caron /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedillar /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def } if % font ISOEncode font % This procedure changes the encoding of a font from the default % Postscript encoding to ISOLatin1. It's typically invoked just % before invoking "setfont". The body of this procedure comes from % Section 5.6.1 of the Postscript book. /ISOEncode { dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end % I'm not sure why it's necessary to use "definefont" on this new % font, but it seems to be important; just use the name "Temporary" % for the font. /Temporary exch definefont } bind def % StrokeClip % % This procedure converts the current path into a clip area under % the assumption of stroking. It's a bit tricky because some Postscript % interpreters get errors during strokepath for dashed lines. If % this happens then turn off dashes and try again. /StrokeClip { {strokepath} stopped { (This Postscript printer gets limitcheck overflows when) = (stippling dashed lines; lines will be printed solid instead.) = [] 0 setdash strokepath} if clip } bind def % desiredSize EvenPixels closestSize % % The procedure below is used for stippling. Given the optimal size % of a dot in a stipple pattern in the current user coordinate system, % compute the closest size that is an exact multiple of the device's % pixel size. This allows stipple patterns to be displayed without % aliasing effects. /EvenPixels { % Compute exact number of device pixels per stipple dot. dup 0 matrix currentmatrix dtransform dup mul exch dup mul add sqrt % Round to an integer, make sure the number is at least 1, and compute % user coord distance corresponding to this. dup round dup 1 lt {pop 1} if exch div mul } bind def % width height string StippleFill -- % % Given a path already set up and a clipping region generated from % it, this procedure will fill the clipping region with a stipple % pattern. "String" contains a proper image description of the % stipple pattern and "width" and "height" give its dimensions. Each % stipple dot is assumed to be about one unit across in the current % user coordinate system. This procedure trashes the graphics state. /StippleFill { % The following code is needed to work around a NeWSprint bug. /tmpstip 1 index def % Change the scaling so that one user unit in user coordinates % corresponds to the size of one stipple dot. 1 EvenPixels dup scale % Compute the bounding box occupied by the path (which is now % the clipping region), and round the lower coordinates down % to the nearest starting point for the stipple pattern. Be % careful about negative numbers, since the rounding works % differently on them. pathbbox 4 2 roll 5 index div dup 0 lt {1 sub} if cvi 5 index mul 4 1 roll 6 index div dup 0 lt {1 sub} if cvi 6 index mul 3 2 roll % Stack now: width height string y1 y2 x1 x2 % Below is a doubly-nested for loop to iterate across this area % in units of the stipple pattern size, going up columns then % across rows, blasting out a stipple-pattern-sized rectangle at % each position 6 index exch { 2 index 5 index 3 index { % Stack now: width height string y1 y2 x y gsave 1 index exch translate 5 index 5 index true matrix tmpstip imagemask grestore } for pop } for pop pop pop pop pop } bind def % -- AdjustColor -- % Given a color value already set for output by the caller, adjusts % that value to a grayscale or mono value if requested by the CL % variable. /AdjustColor { CL 2 lt { currentgray CL 0 eq { .5 lt {0} {1} ifelse } if setgray } if } bind def % x y strings spacing xoffset yoffset justify stipple DrawText -- % This procedure does all of the real work of drawing text. The % color and font must already have been set by the caller, and the % following arguments must be on the stack: % % x, y - Coordinates at which to draw text. % strings - An array of strings, one for each line of the text item, % in order from top to bottom. % spacing - Spacing between lines. % xoffset - Horizontal offset for text bbox relative to x and y: 0 for % nw/w/sw anchor, -0.5 for n/center/s, and -1.0 for ne/e/se. % yoffset - Vertical offset for text bbox relative to x and y: 0 for % nw/n/ne anchor, +0.5 for w/center/e, and +1.0 for sw/s/se. % justify - 0 for left justification, 0.5 for center, 1 for right justify. % stipple - Boolean value indicating whether or not text is to be % drawn in stippled fashion. If text is stippled, % procedure StippleText must have been defined to call % StippleFill in the right way. % % Also, when this procedure is invoked, the color and font must already % have been set for the text. /DrawText { /stipple exch def /justify exch def /yoffset exch def /xoffset exch def /spacing exch def /strings exch def % First scan through all of the text to find the widest line. /lineLength 0 def strings { stringwidth pop dup lineLength gt {/lineLength exch def} {pop} ifelse newpath } forall % Compute the baseline offset and the actual font height. 0 0 moveto (TXygqPZ) false charpath pathbbox dup /baseline exch def exch pop exch sub /height exch def pop newpath % Translate coordinates first so that the origin is at the upper-left % corner of the text's bounding box. Remember that x and y for % positioning are still on the stack. translate lineLength xoffset mul strings length 1 sub spacing mul height add yoffset mul translate % Now use the baseline and justification information to translate so % that the origin is at the baseline and positioning point for the % first line of text. justify lineLength mul baseline neg translate % Iterate over each of the lines to output it. For each line, % compute its width again so it can be properly justified, then % display it. strings { dup stringwidth pop justify neg mul 0 moveto stipple { % The text is stippled, so turn it into a path and print % by calling StippledText, which in turn calls StippleFill. % Unfortunately, many Postscript interpreters will get % overflow errors if we try to do the whole string at % once, so do it a character at a time. gsave /char (X) def { char 0 3 -1 roll put currentpoint gsave char true charpath clip StippleText grestore char stringwidth translate moveto } forall grestore } {show} ifelse 0 spacing neg translate } forall } bind def /CL 1 def save 306.0 396.0 translate 0.9589 0.9589 scale -248 -186 translate 0 373 moveto 496 373 lineto 496 0 lineto 0 0 lineto closepath clip newpath gsave 100 273 moveto 100 228 lineto 55 228 lineto 55 318 lineto 145 318 lineto 145 273 lineto 100 273 lineto 1 setlinewidth 1 setlinecap 1 setlinejoin 0.000 0.000 0.000 setrgbcolor AdjustColor stroke grestore gsave 55 228 moveto 100 228 lineto 100 183 lineto 10 183 lineto 10 273 lineto 55 273 lineto 55 228 lineto 1 setlinewidth 1 setlinecap 1 setlinejoin 0.000 0.000 0.000 setrgbcolor AdjustColor stroke grestore gsave 55 318 moveto 55 273 lineto 10 273 lineto 10 363 lineto 100 363 lineto 100 318 lineto 55 318 lineto 1 setlinewidth 1 setlinecap 1 setlinejoin 0.000 0.000 0.000 setrgbcolor AdjustColor stroke grestore gsave 145 318 moveto 100 318 lineto 100 363 lineto 190 363 lineto 190 273 lineto 145 273 lineto 145 318 lineto 1 setlinewidth 1 setlinecap 1 setlinejoin 0.000 0.000 0.000 setrgbcolor AdjustColor stroke grestore restore showpage end %%EndDocument @endspecial 1246 4331 a(Figure)f(3.)44 b(The)33 b(c)m(hair)h (substitution)1925 5519 y(5)p eop %%Page: 6 7 6 6 bop 510 4778 a @beginspecial 150 @llx 133 @lly 325 @urx 410 @ury 2880 @rwi @setspecial %%BeginDocument: fig4.ps gsave newpath 150 133 moveto 325 133 lineto 325 410 lineto 150 410 lineto closepath clip newpath 50 dict begin % This is a standard prolog for Postscript generated by Tk's canvas % widget. % SCCS: @(#) prolog.ps 1.5 96/02/17 17:45:11 % The definitions below just define all of the variables used in % any of the procedures here. This is needed for obscure reasons % explained on p. 716 of the Postscript manual (Section H.2.7, % "Initializing Variables," in the section on Encapsulated Postscript). /baseline 0 def /stipimage 0 def /height 0 def /justify 0 def /lineLength 0 def /spacing 0 def /stipple 0 def /strings 0 def /xoffset 0 def /yoffset 0 def /tmpstip null def % Define the array ISOLatin1Encoding (which specifies how characters are % encoded for ISO-8859-1 fonts), if it isn't already present (Postscript % level 2 is supposed to define it, but level 1 doesn't). systemdict /ISOLatin1Encoding known not { /ISOLatin1Encoding [ /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /dotlessi /grave /acute /circumflex /tilde /macron /breve /dotaccent /dieresis /space /ring /cedilla /space /hungarumlaut /ogonek /caron /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedillar /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def } if % font ISOEncode font % This procedure changes the encoding of a font from the default % Postscript encoding to ISOLatin1. It's typically invoked just % before invoking "setfont". The body of this procedure comes from % Section 5.6.1 of the Postscript book. /ISOEncode { dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end % I'm not sure why it's necessary to use "definefont" on this new % font, but it seems to be important; just use the name "Temporary" % for the font. /Temporary exch definefont } bind def % StrokeClip % % This procedure converts the current path into a clip area under % the assumption of stroking. It's a bit tricky because some Postscript % interpreters get errors during strokepath for dashed lines. If % this happens then turn off dashes and try again. /StrokeClip { {strokepath} stopped { (This Postscript printer gets limitcheck overflows when) = (stippling dashed lines; lines will be printed solid instead.) = [] 0 setdash strokepath} if clip } bind def % desiredSize EvenPixels closestSize % % The procedure below is used for stippling. Given the optimal size % of a dot in a stipple pattern in the current user coordinate system, % compute the closest size that is an exact multiple of the device's % pixel size. This allows stipple patterns to be displayed without % aliasing effects. /EvenPixels { % Compute exact number of device pixels per stipple dot. dup 0 matrix currentmatrix dtransform dup mul exch dup mul add sqrt % Round to an integer, make sure the number is at least 1, and compute % user coord distance corresponding to this. dup round dup 1 lt {pop 1} if exch div mul } bind def % width height string StippleFill -- % % Given a path already set up and a clipping region generated from % it, this procedure will fill the clipping region with a stipple % pattern. "String" contains a proper image description of the % stipple pattern and "width" and "height" give its dimensions. Each % stipple dot is assumed to be about one unit across in the current % user coordinate system. This procedure trashes the graphics state. /StippleFill { % The following code is needed to work around a NeWSprint bug. /tmpstip 1 index def % Change the scaling so that one user unit in user coordinates % corresponds to the size of one stipple dot. 1 EvenPixels dup scale % Compute the bounding box occupied by the path (which is now % the clipping region), and round the lower coordinates down % to the nearest starting point for the stipple pattern. Be % careful about negative numbers, since the rounding works % differently on them. pathbbox 4 2 roll 5 index div dup 0 lt {1 sub} if cvi 5 index mul 4 1 roll 6 index div dup 0 lt {1 sub} if cvi 6 index mul 3 2 roll % Stack now: width height string y1 y2 x1 x2 % Below is a doubly-nested for loop to iterate across this area % in units of the stipple pattern size, going up columns then % across rows, blasting out a stipple-pattern-sized rectangle at % each position 6 index exch { 2 index 5 index 3 index { % Stack now: width height string y1 y2 x y gsave 1 index exch translate 5 index 5 index true matrix tmpstip imagemask grestore } for pop } for pop pop pop pop pop } bind def % -- AdjustColor -- % Given a color value already set for output by the caller, adjusts % that value to a grayscale or mono value if requested by the CL % variable. /AdjustColor { CL 2 lt { currentgray CL 0 eq { .5 lt {0} {1} ifelse } if setgray } if } bind def % x y strings spacing xoffset yoffset justify stipple DrawText -- % This procedure does all of the real work of drawing text. The % color and font must already have been set by the caller, and the % following arguments must be on the stack: % % x, y - Coordinates at which to draw text. % strings - An array of strings, one for each line of the text item, % in order from top to bottom. % spacing - Spacing between lines. % xoffset - Horizontal offset for text bbox relative to x and y: 0 for % nw/w/sw anchor, -0.5 for n/center/s, and -1.0 for ne/e/se. % yoffset - Vertical offset for text bbox relative to x and y: 0 for % nw/n/ne anchor, +0.5 for w/center/e, and +1.0 for sw/s/se. % justify - 0 for left justification, 0.5 for center, 1 for right justify. % stipple - Boolean value indicating whether or not text is to be % drawn in stippled fashion. If text is stippled, % procedure StippleText must have been defined to call % StippleFill in the right way. % % Also, when this procedure is invoked, the color and font must already % have been set for the text. /DrawText { /stipple exch def /justify exch def /yoffset exch def /xoffset exch def /spacing exch def /strings exch def % First scan through all of the text to find the widest line. /lineLength 0 def strings { stringwidth pop dup lineLength gt {/lineLength exch def} {pop} ifelse newpath } forall % Compute the baseline offset and the actual font height. 0 0 moveto (TXygqPZ) false charpath pathbbox dup /baseline exch def exch pop exch sub /height exch def pop newpath % Translate coordinates first so that the origin is at the upper-left % corner of the text's bounding box. Remember that x and y for % positioning are still on the stack. translate lineLength xoffset mul strings length 1 sub spacing mul height add yoffset mul translate % Now use the baseline and justification information to translate so % that the origin is at the baseline and positioning point for the % first line of text. justify lineLength mul baseline neg translate % Iterate over each of the lines to output it. 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3508 y Fl(suc)m(h)i(that)d(ev)m(ery)i (\014nite)f(subpatc)m(h)i(of)e Fh(x)g Fl(is)h(congruen)m(t)g(to)f(a)g (subpatc)m(h)h(of)g Fh(\036)3138 3471 y Fj(n)3192 3508 y Fl(\()p Fh(T)14 b Fl(\))36 b(for)h(some)150 3627 y Fh(n)e(>)g Fl(0)j(and)g Fh(T)49 b Ff(2)35 b(A)p Fl(.)57 b(W)-8 b(e)38 b(assume)g Fh(x)g Fl(can)g(b)s(e)g(decomp)s(osed)h(in)e (one)h(and)g(only)f(one)h(w)m(a)m(y)g(b)m(y)150 3747 y(tiles)33 b(of)g(lev)m(el)g Fh(n)p Fl(,)g(for)g(an)m(y)h(\014xed)f Fh(n)28 b Ff(\025)g Fl(0)33 b([Sol].)349 3926 y(The)44 b(ab)s(o)m(v)m(e)f(de\014nition)h(giv)m(es)f(spaces)h(of)g(tilings)e (in)h(whic)m(h)h(the)g(tiles)f(can)g(app)s(ear)h(in)150 4046 y(arbitrary)27 b(orien)m(tations.)42 b(Although)27 b(an)m(y)h Fd(\014xe)-5 b(d)28 b Fl(c)m(hair)g(tiling,)f(for)i (example,)f(has)g(tiles)g(in)g(only)150 4165 y(four)j(orien)m(tations,) g(the)g(space)h(of)f Fd(al)5 b(l)31 b Fl(c)m(hair)g(tilings)f(also)h (con)m(tains)g(rotated)f(v)m(ersions)h(of)g(that)150 4285 y(tiling,)h(and)i(so)f(con)m(tains)h(tilings)e(in)h(whic)m(h)h(c)m (hairs)g(app)s(ear)g(in)f(an)m(y)g(orien)m(tation.)349 4404 y(In)43 b(man)m(y)f(cases,)k(esp)s(ecially)c(when)h(w)m(orking)f (with)g(tilings)f(whose)i(tiles)g(ha)m(v)m(e)f(only)g(a)150 4524 y(\014nite)d(n)m(um)m(b)s(er)h(of)f(orien)m(tations)f(p)s(er)h (tiling,)g(it)f(is)h(con)m(v)m(enien)m(t)h(\(and)e(customary\))h(to)f (allo)m(w)150 4643 y(tiles)d(only)g(a)h(minimal)f(set)g(of)h(orien)m (tations.)51 b(Instead)37 b(of)f(using)g(the)g(metric)f Fh(d)p Fl(,)h(w)m(e)g(de\014ne)h(a)150 4763 y(metric)f Fh(d)515 4727 y Fe(0)578 4763 y Fl(suc)m(h)h(that)e(t)m(w)m(o)g (tilings)g(are)h(close)g(if)f(\(and)h(only)f(if)8 b(\))35 b(they)g(agree)h(on)g(a)f(large)g(disk)150 4883 y(around)41 b(the)g(origin)f Fd(up)j(to)f(a)h(smal)5 b(l)43 b(tr)-5 b(anslation)p Fl(,)42 b(rather)f(than)g(up)g(to)f(a)g(small)g (Euclidean)150 5002 y(motion.)j(Sp)s(eci\014cally)-8 b(,)632 5243 y Fh(d)684 5202 y Fe(0)712 5243 y Fl(\()p Fh(x;)17 b(x)910 5202 y Fe(0)937 5243 y Fl(\))27 b(=)h(inf)9 b Ff(f)1377 5176 y Fh(R)23 b Fl(+)f Fh(S)p 1292 5220 437 4 v 1292 5312 a Fl(1)f(+)i Fh(R)g Fl(+)f Fh(S)1768 5243 y Fl(:)27 b Fh(B)1899 5261 y Fk(1+1)p Fj(=S)2138 5243 y Fl(\()p Fh(g)t Fl([)p Fh(@)6 b(x)p Fl(]\))25 b(=)j Fh(B)2646 5261 y Fk(1+1)p Fj(=S)2884 5243 y Fl(\()p Fh(h)p Fl([)p Fh(@)6 b(x)3124 5202 y Fe(0)3152 5243 y Fl(]\))p Ff(g)364 b Fl(2)3683 5207 y Fe(0)3711 5243 y Fl(\))1900 5519 y(11)p eop %%Page: 12 13 12 12 bop 150 100 a Fl(where)50 b Fh(g)t(;)17 b(h)53 b Ff(2)i Fg(R)854 63 y Fk(2)898 100 y Fl(,)e Fh(R)i(>)f Fl(0)p Fh(;)65 b(S)60 b(>)55 b Fl(0)p Fh(;)65 b Ff(j)p Fh(g)t Ff(j)53 b Fh(<)i(R)q(;)64 b Ff(j)p Fh(h)p Ff(j)54 b Fh(<)h(R)50 b Fl(and)f Fh(d)2928 63 y Fe(0)2956 100 y Fl(\()p Fh(x;)17 b(x)3154 63 y Fe(0)3181 100 y Fl(\))54 b Ff(\021)g Fl(1)49 b(if)h(the)150 219 y(de\014ning)39 b(set)f(is)g(empt)m(y)-8 b(.)58 b(W)-8 b(e)38 b(pic)m(k)g(an)m(y)f(one) i(tiling)d Fh(x)g Ff(2)f Fh(X)2457 234 y Fj(\036)2510 219 y Fl(,)j(and)h(de\014ne)g Fh(X)3150 234 y Fj(x)3235 219 y Ff(\032)d Fh(X)3431 234 y Fj(\036)3521 219 y Fl(to)i(b)s(e)150 339 y Ff(f)p Fh(x)257 303 y Fe(0)312 339 y Ff(2)28 b Fh(X)489 354 y Fj(\036)570 339 y Fl(:)f Fh(d)677 303 y Fe(0)705 339 y Fl(\()p Fh(x;)17 b(x)903 303 y Fe(0)930 339 y Fl(\))27 b Fh(<)h Fl(1)p Ff(g)p Fl(.)42 b(F)-8 b(or)29 b(tiling)f(spaces)i(with)f(only)f(a)h(\014nite)g(n)m(um)m(b)s (er)i(of)e(orien)m(tations)150 458 y(p)s(er)34 b(tiling,)e Fh(X)696 473 y Fj(x)778 458 y Fl(consists)i(of)g(those)g(tilings)e(in)h (whic)m(h)i(tiles)d(only)h(app)s(ear)h(parallel)e(to)h(tiles)g(in)150 578 y Fh(x)p Fl(.)44 b(W)-8 b(e)33 b(call)g Fh(X)716 593 y Fj(x)799 578 y Fl(a)g(\\\014xed-orien)m(tation")g(tiling)f (space.)349 697 y(Finally)-8 b(,)24 b(w)m(e)g(consider)h(the)g(quotien) m(t)e(of)h Fh(X)1949 712 y Fj(\036)2026 697 y Fl(b)m(y)g(rotations)f (ab)s(out)g(the)i(origin)e(and)h(denote)150 817 y(this)31 b(space)i Fh(X)690 832 y Fk(0)734 817 y Fl(.)43 b(F)-8 b(or)31 b(an)m(y)g(\014xed)h(tiling)e Fh(x)p Fl(,)i Fh(X)1865 832 y Fk(0)1940 817 y Fl(is)g(also)e(the)i(quotien)m(t)f(of)h Fh(X)2992 832 y Fj(x)3073 817 y Fl(b)m(y)f(the)h(\\relativ)m(e)150 936 y(orien)m(tation)47 b(group")g Fh(G)g Fl(of)h(the)f(tiling)f Fh(x)p Fl(,)51 b(de\014ned)e(as)e(follo)m(ws:)72 b Fh(G)47 b Fl(is)g(generated)h(b)m(y)g(the)150 1056 y(rotations)34 b(that)h(tak)m(e)g(a)g(tile)f(in)h Fh(x)g Fl(and)h(turn)f(it)g (parallel)f(to)h(another)g(tile)g(of)g(the)h(same)f(t)m(yp)s(e)150 1176 y(in)f Fh(x)p Fl(.)44 b(In)34 b([RaS])g(it)f(w)m(as)g(sho)m(wn)i (that)e(this)h(group)g(is)f(the)h(same)g(for)g(all)f(tilings)f Fh(x)c Ff(2)h Fh(X)3453 1191 y Fj(\036)3505 1176 y Fl(.)45 b(The)150 1295 y(top)s(ologies)30 b(of)h(these)h(three)g(spaces)g(are)f (closely)g(related.)43 b(The)31 b(space)h Fh(X)2936 1310 y Fj(\036)3020 1295 y Fl(is)f(a)f(circle)i(bundle)150 1415 y(o)m(v)m(er)g Fh(X)445 1430 y Fk(0)521 1415 y Fl(with)f(some)h (singular)f(\014b)s(ers;)i(these)g(\014b)s(ers)f(corresp)s(ond)h(to)f (tilings)e(with)i(a)f(discrete)150 1534 y(rotational)h(symmetry)g(ab)s (out)h(the)h(origin.)349 1654 y(Next)27 b(w)m(e)h(sho)m(w)g(ho)m(w)f (to)g(giv)m(e)g(a)g(substitution)h(tiling)e(space)j(the)f(structure)g (of)g(an)f(in)m(v)m(erse)150 1773 y(limit.)62 b(W)-8 b(e)40 b(b)s(egin)g(b)m(y)g(doing)g(this)f(\014rst)h(for)g(a)g (\014xed-orien)m(tation)g(space,)i(and)e(w)m(e)g(b)s(egin)g(in)150 1893 y(particular)47 b(with)h(the)g(example)g(of)f(the)h(c)m(hair.)88 b(Fix)47 b(a)h(c)m(hair)g(tiling)e Fh(x)p Fl(,)51 b(and)d(in)g(it)f(a)h (tile)150 2012 y Fh(T)14 b Fl(.)77 b(Eac)m(h)45 b(of)f(the)h(other)g (tiles)e(in)i Fh(x)f Fl(are)g(rotated)g(with)g(resp)s(ect)i(to)d Fh(T)59 b Fl(b)m(y)44 b(an)h(elemen)m(t)g(of)150 2132 y(the)40 b(cyclic)f(rotation)g(group)h Fh(G)1364 2147 y Fj(ch)1489 2132 y Fl(=)f Fg(Z)1671 2147 y Fk(4)1716 2132 y Fl(.)63 b(W)-8 b(e)40 b(construct)h(an)e(\(uncollared\))h(c)m (hair)g(complex)150 2252 y(\006)222 2215 y Fj(uc)351 2252 y Fl(as)i(follo)m(ws.)68 b(W)-8 b(e)42 b(start)f(with)g(the)h(top) s(ological)e(disjoin)m(t)i(sum)g(\006)2883 2215 y Fe(0)2953 2252 y Fl(of)f(the)h(elemen)m(ts)h(of)150 2371 y Ff(f)p Fh(g)t Fl([)p Fh(T)14 b Fl(])46 b(:)i Fh(g)34 b Ff(2)e Fh(G)760 2386 y Fj(ch)847 2371 y Ff(g)p Fl(.)50 b(W)-8 b(e)36 b(then)g(iden)m(tify)g(those)g(edges)g(in)g(\006)2465 2335 y Fe(0)2528 2371 y Fl(whic)m(h)g(\\meet")f(somewhere)i(in)150 2491 y Fh(x)p Fl(,)d(de\014ning)h(\006)718 2455 y Fj(uc)805 2491 y Fl(.)46 b(W)-8 b(e)34 b(consider)h(a)f(coun)m(table)g(n)m(um)m (b)s(er)i(of)e(copies)g(\006)2819 2455 y Fj(uc)2819 2517 y(j)2941 2491 y Fl(of)g(\006)3127 2455 y Fj(uc)3215 2491 y Fl(,)f(indexed)i(b)m(y)150 2610 y(the)h(non-negativ)m(e)g(in)m (tegers.)51 b(W)-8 b(e)35 b(think)g(of)h(the)g(tiles)f(in)g(\006)2458 2574 y Fj(uc)2458 2636 y(j)2581 2610 y Fl(as)g(referring)i(to)e(lev)m (el)g Fh(j)41 b Fl(tiles.)150 2730 y(The)31 b(substitution)g Fh(\036)g Fl(\(whereb)m(y)g(w)m(e)g(think)g(of)f(eac)m(h)i(tile)e(of)h (lev)m(el)f Fh(j)22 b Fl(+)c(1)30 b(as)h(the)g(union)g(of)f(four)150 2849 y(tiles)36 b(of)g(lev)m(el)g Fh(j)6 b Fl(\))35 b(then)i(de\014nes) h(a)e(map)g(from)g(\006)2011 2813 y Fj(uc)2011 2875 y(j)t Fk(+1)2190 2849 y Fl(to)f(\006)2386 2813 y Fj(uc)2386 2875 y(j)2510 2849 y Fl(for)h(an)m(y)g Fh(j)i Ff(\025)33 b Fl(0,)j(and)h(allo)m(ws)e(us)150 2969 y(to)e(de\014ne)i(the)e(in)m(v) m(erse)h(limit)1303 2978 y(\006)1299 3038 y Fe( )1379 2933 y Fj(uc)1494 2969 y Ff(\021)28 b Fl(lim)1604 3029 y Fe( )-14 b(\000)1754 2969 y Fl(\006)1826 2933 y Fj(uc)1826 2995 y(j)1914 2969 y Fl(.)349 3117 y(There)35 b(is)f(a)f(map)h(from)h (the)f(tiling)f(space)h Fh(X)2063 3132 y Fj(x)2147 3117 y Fl(to)2273 3126 y(\006)2269 3186 y Fe( )2349 3081 y Fj(uc)2470 3117 y Fl(de\014ned,)i(on)e Fh(x)29 b Ff(2)g Fh(X)3245 3132 y Fj(x)3294 3117 y Fl(,)34 b(b)m(y)g(deter-)150 3265 y(mining)c(precisely)h(ho)m(w)g(the)g(origin)e(in)i Fh(x)f Fl(sits)g(in)h(eac)m(h)g(successiv)m(e)h(tile)e(of)h(higher)g (and)g(higher)150 3385 y(lev)m(el.)47 b(F)-8 b(or)34 b(man)m(y)g(but)h(not)f(all)g(substitution)h(systems,)f(this)h(map)f (is)h(neither)g(1-1)f(nor)g(on)m(to.)150 3504 y(A)c(p)s(oin)m(t)g(in)h (the)f(in)m(v)m(erse)i(limit)d(precisely)h(describ)s(es)i(the)f (hierarc)m(h)m(y)g(of)g(tilies)e(con)m(taining)i(the)150 3624 y(origin.)47 b(In)35 b(some)g(instances)h(this)e(union)i(of)e (tiles)h(ma)m(y)f(not)g(co)m(v)m(er)h(the)g(en)m(tire)g(plane;)h(if)e (not,)150 3743 y(it)h(ma)m(y)g(b)s(e)h(that)f(there)h(is)f(more)g(than) h(one)g(w)m(a)m(y)f(to)g(extend)h(the)g(tiling)e(to)h(co)m(v)m(er)h (the)f(en)m(tire)150 3863 y(plane)45 b(\(in)f(whic)m(h)i(case)f(the)g (map)g(is)f(not)h(1-1\),)i(or)d(it)g(ma)m(y)g(b)s(e)h(that)f(there)i (is)e(no)h(w)m(a)m(y)f(to)150 3982 y(extend)30 b(it)f(\(in)g(whic)m(h)i (case)f(the)g(map)g(is)f(not)h(on)m(to\).)42 b(F)-8 b(or)29 b(example,)h(in)g(a)f(c)m(hair)h(tiling)f(the)h(tile)150 4102 y(con)m(taining)e(the)h(origin)f(migh)m(t)g(sit)g(in)g(the)h(upp)s (er)g(left)f(corner)h(of)g(the)f(tile)g(of)h(lev)m(el)e(1)i(of)f(Fig.)g (2,)150 4221 y(whic)m(h)39 b(sits)f(in)h(the)f(same)h(p)s(osition)f(in) g(a)g(corresp)s(onding)i(tile)d(of)i(lev)m(el)f(2,)h(and)g(so)f(on.)59 b(The)150 4341 y(union)32 b(of)f(these)h(tiles)f(co)m(v)m(ers)h(only)e (a)h(quadran)m(t,)h(and)f(this)h(can)f(b)s(e)h(extended)g(to)f(a)g (complete)150 4460 y(c)m(hair)j(tiling)e(of)h(the)h(plane)f(in)h(more)f (than)g(one)h(w)m(a)m(y)-8 b(.)349 4580 y(This)46 b(problem)g(is)g (absen)m(t)h(in)f(substitution)g(systems)g(\(suc)m(h)h(as)f(the)g(P)m (enrose)h(tilings)150 4700 y([AnP]\))39 b(whic)m(h)i(\\force)g(the)f(b) s(order",)i(that)e(is,)h(for)g(whic)m(h)g(there)f(is)g(some)h(in)m (teger)f Fh(N)51 b Fl(suc)m(h)150 4819 y(that,)30 b(in)g(ev)m(ery)g (tiling,)g(for)g(ev)m(ery)g(tile)g Fh(\036)1687 4783 y Fj(N)1763 4819 y Fh(T)44 b Fl(of)30 b(lev)m(el)g Fh(N)41 b Fl(the)31 b(\(lev)m(el)e(0\))h(tiles)g(whic)m(h)h(abut)f(this)150 4939 y(tile)g(are)g(completely)g(determined)i(b)m(y)f(the)f(t)m(yp)s(e) h Fh(T)14 b Fl(.)43 b(F)-8 b(rom)30 b(this)g(it)g(follo)m(ws)g(that,)h (for)f(an)m(y)h Fh(M)11 b Fl(,)150 5058 y(the)35 b(tiles)f(of)h(lev)m (el)f Fh(M)45 b Fl(that)34 b(abut)h(a)f(giv)m(en)g(tile)g(of)h(lev)m (el)f Fh(N)g Fl(+)23 b Fh(M)45 b Fl(are)35 b(also)f(determined.)49 b(An)150 5178 y(in\014nite-lev)m(el)30 b(tile,)g(therefore,)h (determines)g(a)e(tiling)g(of)h(the)g(en)m(tire)g(plane,)h(and)f(is)g (consisten)m(t)1900 5519 y(12)p eop %%Page: 13 14 13 13 bop 150 100 a Fl(with)28 b(a)h(tiling)e(of)i(the)g(en)m(tire)g (plane,)h(ev)m(en)f(if)g(it)f(do)s(es)h(not)g(itself)f(co)m(v)m(er)h (the)g(en)m(tire)h(plane.)42 b(The)150 219 y(map)33 b(from)h(the)f (space)i(of)e(tilings)g(to)f(the)i(in)m(v)m(erse)g(limit)e(is)h (therefore)i(1-1)e(and)g(on)m(to.)349 339 y(It)40 b(is)h(easy)f(to)g (pro)m(v)m(e,)i(but)f(v)m(ery)f(useful,)j(that)d(for)g(an)m(y)h 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b(the)g(complex)g(\006)2009 1942 y Fk(0)2091 1927 y Fl(the)g(basic)g(cells)g(are)g(tiles)f Fh(T)14 b Fl(,)39 b(where)f(eac)m(h)150 2047 y(t)m(yp)s(e)29 b(app)s(ears)g(in)g(only)f(one)h(orien)m(tation.)42 b(As)29 b(b)s(efore,)i(if)d(it)h(is)f(p)s(ossible)i(for)f(a)f(tile)h Fh(T)3349 2062 y Fk(1)3422 2047 y Fl(to)g(meet)150 2166 y(a)k(tile)f Fh(T)462 2181 y Fk(2)539 2166 y Fl(then)i(their)f(common)g (edge)g(is)g(iden)m(ti\014ed,)h(and)f(it)f(ma)m(y)h(b)s(e)g(necessary)h (to)e(consider)150 2286 y(collared)38 b(tiles.)57 b(As)38 b(b)s(efore)h(the)f(substitution)g Fh(\036)g Fl(maps)g(\006)2414 2301 y Fk(0)2497 2286 y Fl(to)f(itself)h(and)g(w)m(e)h(consider)g(the) 150 2405 y(in)m(v)m(erse)33 b(limit.)42 b(A)31 b(p)s(oin)m(t)h(in)g (the)g(in)m(v)m(erse)g(limit)f(is)g(a)h(consisten)m(t)g(instruction:)44 b(the)32 b(origin)f(sits)150 2525 y(at)36 b(suc)m(h-and-suc)m(h)k(a)c (p)s(oin)m(t)g(in)g(suc)m(h-and-suc)m(h)k(a)c(tile,)g(whic)m(h)i(sits)e (in)g(a)g(particular)g(w)m(a)m(y)g(in)150 2645 y(a)e(tile)g(of)h(lev)m (el)f(1,)g(whic)m(h)h(sits)g(in)f(a)h(particular)f(w)m(a)m(y)g(in)h(a)f (tile)g(of)g(lev)m(el)g(2,)h(and)g(so)f(on.)48 b(Since)150 2764 y(w)m(e)35 b(are)g(using)h(collared)e(tiles)h(\(or)f(tiles)h(that) f(force)i(the)f(b)s(order\),)g(this)g(prescription)h(de\014nes)150 2884 y(a)d(unique)h(tiling)e(of)h(the)h(en)m(tire)g(plane,)f(up)h(to)f (an)g(o)m(v)m(erall)g(rotation,)f(i.e.)43 b(a)33 b(p)s(oin)m(t)g(in)h Fh(X)3522 2899 y Fk(0)3566 2884 y Fl(.)349 3003 y(The)49 b(construction)g(of)g(\006)1356 3018 y Fj(\036)1457 3003 y Fl(is)f(similar,)k(except)c(that)g(the)h(basic)g(cells)g(are)f(pro)s (ducts)150 3123 y Fh(S)217 3087 y Fk(1)287 3123 y Ff(\002)26 b Fh(T)14 b Fl(,)40 b(where)f(the)g Fh(T)52 b Fl(are)39 b(as)f(in)h(the)g(construction)g(of)g(\006)2475 3138 y Fk(0)2519 3123 y Fl(.)60 b(If)39 b(a)f(tile)g Fh(T)3035 3138 y Fk(1)3080 3123 y Fl(,)h(rotated)f(b)m(y)h(an)150 3242 y(angle)31 b Fh(\013)g Fl(\(with)f(resp)s(ect)i(to)f(the)g 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b(these)j(complexes)e(are)h(closely)e(related.)54 b(Just)36 b(as)h Fh(X)2747 5051 y Fk(0)2827 5036 y Fl(is)f(the)h(quotien)m(t)f (of)g Fh(X)3700 5051 y Fj(x)150 5155 y Fl(b)m(y)30 b Fh(G)h Fl(and)f(the)h(quotien)m(t)e(of)i Fh(X)1336 5170 y Fj(\036)1418 5155 y Fl(b)m(y)f Fh(S)1620 5119 y Fk(1)1665 5155 y Fl(,)g(\006)1795 5170 y Fk(0)1870 5155 y Fl(is)g(the)g(quotien)m (t)g(of)h(\006)2708 5170 y Fj(x)2788 5155 y Fl(b)m(y)f Fh(G)g Fl(and)h(the)f(quotien)m(t)150 5275 y(of)j(\006)335 5290 y Fj(\036)422 5275 y Fl(b)m(y)g Fh(S)627 5239 y Fk(1)671 5275 y Fl(.)1900 5519 y(13)p eop %%Page: 14 15 14 14 bop 150 100 a Fm(3.)51 b(Finite)37 b(relativ)m(e)f(orien)m (tation)g(group)j(and)e(P-p)s(ositivit)m(y)-10 b(.)349 219 y Fl(Here)33 b(w)m(e)g(de\014ne)i(the)e(\\P-p)s(ositiv)m(e)f(cone") h(of)g(the)g(top)f(dimensional)3003 194 y(\024)2992 219 y(Cec)m(h)i(cohomology)150 339 y(group)42 b(for)g(substitution)g (tiling)f(spaces.)71 b(In)42 b(this)g(section)g(w)m(e)g(assume)h(our)f (system)g(has)g(a)150 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Fl(3\))150 2291 y(for)30 b(all)g Fh(g)j Fl(and)e(all)e Fh(k)s Fl(,)h(where)h(\()p Fh(g)t(;)17 b(k)s Fl(\))28 b(denotes)k(the)e(elemen)m(t)h Fh(g)i Fl(in)e(the)f Fh(k)s Fl(-th)h(cop)m(y)f(of)g(the)h(group.)150 2411 y(The)37 b(direct)g(limit)e(of)i(the)g(cohomology)e(groups)i Fh(H)2141 2375 y Fk(2)2186 2411 y Fl(\(\006)2297 2426 y Fj(x)2347 2411 y Fl(\))f(can)h(b)s(e)g(obtained)g(b)m(y)f(taking)g (the)150 2530 y(direct)44 b(limit)e(of)i(the)g(co)s(c)m(hain)h(groups)f Fh(C)1778 2494 y Fk(2)1823 2530 y Fl(\(\006)1934 2545 y Fj(x)1984 2530 y Fl(\))f(then)i(mo)s(ding)e(out)h(b)m(y)f(the)h (image)g(of)g(the)150 2650 y(cob)s(oundary)e(map)f Fh(\016)46 b Fl(from)41 b(the)h(direct)g(limit)e(of)h Fh(C)2179 2614 y Fk(1)2224 2650 y Fl(\(\006)2335 2665 y Fj(x)2385 2650 y Fl(\).)68 b(\(The)42 b(pro)s(of)f(of)h(this)f(is)h(simple)150 2769 y(diagram)33 b(c)m(hasing.\))349 2889 y(Since)j(\006)683 2904 y Fj(x)768 2889 y Fl(is)e(a)h(\014nite)g(simplicial)f(complex,)h Fh(C)2146 2904 y Fk(2)2190 2889 y Fl(\(\006)2301 2904 y Fj(x)2351 2889 y Fl(\),)g(the)g(space)h(of)f(simplicial)e(c)m(hains,) 150 3009 y(is)e(generated)h(b)m(y)g(the)f(tiles)g(themselv)m(es)h(and)g (is)f(isomorphic)g(to)g Fg(Z)2690 2972 y Fj(n)2776 3009 y Fl(where)h Fh(n)f Fl(is)g(the)g(n)m(um)m(b)s(er)150 3128 y(of)44 b(distinct)g(tiles)g(in)g(\006)1066 3143 y Fj(x)1116 3128 y Fl(.)76 b(T)-8 b(o)44 b(compute)g(the)h(cohomology) -8 b(,)45 b(eac)m(h)g(tile)e(m)m(ust)i(b)s(e)f(giv)m(en)g(an)150 3248 y(orien)m(tation.)53 b(W)-8 b(e)37 b(de\014ne)h(this)f(orien)m (tation)e(in)i(the)g(follo)m(wing)f(w)m(a)m(y:)50 b(Select)37 b(an)f(orien)m(tation)150 3367 y(for)h Fg(R)378 3331 y Fk(2)460 3367 y Fl(and)g(select)h(a)e(tiling)g Fh(x)1340 3382 y Fk(0)1419 3367 y Ff(2)e Fh(X)1602 3382 y Fj(x)1652 3367 y Fl(.)55 b(The)37 b(translations)g(comprise)h(a)e(free)i Fg(R)3264 3331 y Fk(2)3309 3367 y Fl(-action)f(on)150 3487 y Fh(x)207 3502 y Fk(0)252 3487 y Fl(.)42 b(Eac)m(h)28 b(tile)f(that)h(app)s(ears)g(in)g Fh(x)1472 3502 y Fk(0)1544 3487 y Fl(th)m(us)h(inherits)f(an)g(orien)m(tation)f(from)h(the)h (orien)m(tation)e(on)150 3606 y Fg(R)222 3570 y Fk(2)267 3606 y Fl(.)43 b(All)32 b(suc)m(h)i(orien)m(tations)e(are)g(consisten)m (t)i(and)f(are)f(indep)s(enden)m(t)j(of)d Fh(x)2989 3621 y Fk(0)3034 3606 y Fl(.)44 b(The)33 b(pro)6 b(jection)150 3726 y(map)31 b(from)g Fh(X)685 3741 y Fj(x)765 3726 y Fl(to)f(\006)956 3741 y Fj(x)1037 3726 y Fl(de\014nes)j(an)e(orien)m (tation)f(on)h(tiles)f(in)h(\006)2532 3741 y Fj(x)2582 3726 y Fl(.)43 b(F)-8 b(rom)30 b(here)i(on,)f(w)m(e)g(refer)g(to)150 3845 y(this)i(as)h(the)f(\\p)s(ositiv)m(e)g(orien)m(tation")f(of)i(the) f(tiles)g(in)g(\006)2297 3860 y Fj(x)2347 3845 y Fl(.)349 3965 y(If)49 b(all)e(tiles)g(at)g(all)g(lev)m(els)h(are)g(giv)m(en)g (the)g(p)s(ositiv)m(e)f(orien)m(tation,)k(the)d(induced)i(map)150 4084 y Fh(\036)209 4099 y Fe(\003)283 4084 y Fl(:)27 b Fh(C)409 4099 y Fk(2)454 4084 y Fl(\(\006)565 4099 y Fj(x)615 4084 y Fl(\))g Ff(!)h Fh(C)880 4099 y Fk(2)925 4084 y Fl(\(\006)1036 4099 y Fj(x)1085 4084 y Fl(\))33 b(is)g(an)g Fh(n)22 b Ff(\002)g Fh(n)32 b Fl(matrix)g(with)h(only)f (non-negativ)m(e)i(en)m(tries.)44 b(\(The)34 b Fh(i;)17 b(j)150 4204 y Fl(en)m(try)34 b(in)h(the)f(matrix)f(is)i(the)f(n)m(um)m (b)s(er)i(of)e(copies)h(of)g(tile)f(t)m(yp)s(e)g Fh(i)g Fl(whic)m(h)h(app)s(ear)g(in)f(a)g(lev)m(el)g(1)150 4324 y(tile)27 b(of)h(t)m(yp)s(e)g Fh(j)6 b Fl(.\))42 b(By)27 b(assumption)h(the)g(matrix)f(is)h(primitiv)m(e,)g(that)f(is,)i(some)f (p)s(o)m(w)m(er)g(has)h(all)e(of)150 4443 y(its)k(en)m(tries)h (strictly)f(p)s(ositiv)m(e.)42 b(The)32 b(co)s(c)m(hain)g(group)g Fh(C)2270 4407 y Fk(2)2315 4443 y Fl(\(\006)2426 4458 y Fj(x)2476 4443 y Fl(\))f(is)g(generated)h(b)m(y)g(the)g(duals)f(to) 150 4563 y(the)j(tiles.)45 b(Relativ)m(e)33 b(to)g(this)h(basis,)g(the) g(pullbac)m(k)g(map)g Fh(\036)2404 4527 y Fe(\003)2484 4563 y Fl(is)f(describ)s(ed)i(b)m(y)f(the)g(transp)s(ose)150 4682 y(of)28 b Fh(\036)317 4697 y Fe(\003)363 4682 y Fl(.)42 b(Similar)27 b(constructions)i(apply)f(to)f Fh(C)1839 4646 y Fk(1)1884 4682 y Fl(,)h(except)h(that)e(the)h(elemen)m(ts)h(of)f Fh(C)3205 4697 y Fk(1)3278 4682 y Fl(and)g Fh(C)3544 4646 y Fk(1)3617 4682 y Fl(are)150 4802 y(not)i(naturally)f(orien)m (ted,)i(so)f(the)g(substitution)h(matrix)e(on)h Fh(C)2533 4766 y Fk(1)2608 4802 y Fl(ma)m(y)f(ha)m(v)m(e)i(negativ)m(e)e(en)m (tries.)349 4921 y(Let)39 b Ff(C)45 b Fl(denote)39 b(the)g(direct)g (limit)e(of)i(the)g(co)s(c)m(hains)g Fh(C)2442 4885 y Fk(2)2487 4921 y Fl(\(\006)2598 4936 y Fj(x)2648 4921 y Fl(\))f(under)i(the)f(map)g Fh(\036)3476 4885 y Fe(\003)3522 4921 y Fl(,)g(and)150 5041 y(de\014ne)d Ff(C)491 5056 y Fk(+)593 5041 y Fl(to)e(b)s(e)h(the)g(semigroup)g(of)g(elemen)m(ts)h (\()p Fh(c;)17 b(k)s Fl(\))29 b Ff(2)i(C)41 b Fl(whic)m(h)35 b(ha)m(v)m(e)g(the)g(prop)s(ert)m(y)g(that)150 5160 y(\()p Fh(\036)248 5124 y Fe(\003)294 5160 y Fl(\))333 5124 y Fj(m)408 5160 y Fh(c)40 b Ff(2)f Fl(\()p Fg(Z)701 5175 y Fk(+)768 5160 y Fl(\))807 5124 y Fj(n)901 5160 y Fl(for)h(some)h Fh(m)p Fl(,)h(where)f Fh(n)f Fl(is)g(the)h(n)m(um)m(b)s(er)g(of)g (distinct)f(tiles)g(in)g(\006)3425 5175 y Fj(x)3475 5160 y Fl(.)65 b(It)40 b(is)150 5280 y(clear)34 b(that)g Ff(C)658 5295 y Fk(+)748 5280 y Ff(\000)23 b(C)900 5295 y Fk(+)996 5280 y Fl(=)31 b Ff(C)6 b Fl(.)47 b(Since)35 b(the)g(matrix)e Fh(\036)2055 5244 y Fe(\003)2136 5280 y Fl(con)m(tains)i(only)f (non-negativ)m(e)g(en)m(tries,)i(it)1900 5519 y(14)p eop %%Page: 15 16 15 15 bop 150 100 a Fl(is)39 b(also)g(clear)g(that)g Ff(C)980 115 y Fk(+)1073 100 y Fl(+)26 b Ff(C)1228 115 y Fk(+)1333 100 y Ff(\032)38 b(C)1500 115 y Fk(+)1566 100 y Fl(,)j Ff(C)1687 115 y Fk(+)1780 100 y Ff(\\)26 b(\000C)2001 115 y Fk(+)2106 100 y Fl(=)38 b Ff(f)p Fl(0)p Ff(g)h Fl(and)g(th)m(us)i(\()p Ff(C)6 b Fh(;)17 b Ff(C)3029 115 y Fk(+)3094 100 y Fl(\))39 b(is)h(an)f(ordered)150 219 y(group)34 b(\(in)f(fact,)g(a)g(dimension)h(group)f([Ell]\).)349 339 y(Let)j Fh(H)i Fl(=)31 b Fh(H)850 303 y Fk(2)895 339 y Fl(\()p Fh(X)1017 354 y Fj(x)1066 339 y Fl(\))k(denote)h(the)f (top)1826 314 y(\024)1815 339 y(Cec)m(h)h(cohomology)e(group)i(for)f Fh(X)3132 354 y Fj(x)3181 339 y Fl(.)50 b(W)-8 b(e)35 b(wish)g(to)150 458 y(de\014ne)g(a)e(p)s(ositiv)m(e)f(cone)i(on)g Fh(H)41 b Fl(b)m(y)1521 700 y Fh(H)1604 715 y Fk(+)1698 700 y Fl(=)28 b Ff(f)p Fl([)p Fh(c)p Fl(])33 b Ff(j)f Fh(c)c Ff(2)g(C)2262 715 y Fk(+)2329 700 y Ff(g)1282 b Fl(4\))150 941 y(where)29 b(elemen)m(ts)h(of)f Ff(C)35 b Fl(are)28 b(equiv)-6 b(alen)m(t)29 b(if)g(their)f(di\013erence)i(is)f (in)f(Im)17 b Fh(\016)t Fl(.)42 b(T)-8 b(o)28 b(see)i(that)e(\()p Fh(H)s(;)17 b(H)3647 956 y Fk(+)3711 941 y Fl(\))150 1061 y(is)33 b(an)h(ordered)g(group,)f(w)m(e)h(in)m(tro)s(duce)g(a)f (function)h Fh(\026)28 b Fl(:)f Ff(C)34 b(!)27 b Fg(R)p Fl(.)150 1240 y Fm(Lemma)40 b(1)p Fl(.)47 b Fd(Ther)-5 b(e)37 b(is)f(a)g(homomorphism)h Fh(\026)30 b Fl(:)f Ff(C)36 b(!)29 b Fg(R)37 b Fd(such)f(that)h Ff(C)2830 1255 y Fk(+)2919 1240 y Ff(\000)24 b(f)p Fl(0)p Ff(g)29 b Fl(=)h Fh(\026)3366 1204 y Fe(\000)p Fk(1)3473 1240 y Fl(\()p Fg(R)3584 1255 y Fk(+)3673 1240 y Ff(\000)150 1360 y(f)p Fl(0)p Ff(g)p Fl(\))p Fd(.)150 1539 y Fl(Pro)s(of:)45 b(Since)35 b(the)g(matrix)e Fh(\036)1281 1503 y Fe(\003)1361 1539 y Fl(is)h(primitiv)m(e)g(it)f(has)i(a)f(P)m(erron)g(eigen)m(v)-6 b(alue)35 b Fh(\025)29 b(>)h Fl(0)k(\(of)g(m)m(ulti-)150 1659 y(plicit)m(y)g(1\).)50 b(This)35 b(is)h(the)f(area)g(stretc)m (hing)h(factor)g(of)f(the)h(substitution.)50 b(\(T)-8 b(o)34 b(see)i(this,)g(note)150 1779 y(that)30 b(a)g(v)m(ector)g(whose) h(en)m(tries)f(are)h(the)f(areas)h(of)f(the)g(v)-6 b(arious)30 b(tiles)g(is)g(an)h(eigen)m(v)m(ector)f(of)h(the)150 1898 y(substitution)38 b(matrix.)56 b(Since)38 b(all)f(en)m(tries)h(of) g(this)f(v)m(ector)h(are)g(p)s(ositiv)m(e)f(this)g(is)h(the)f(P)m (erron)150 2018 y(eigen)m(v)m(ector,)g(and)f(the)h(corresp)s(onding)g (eigen)m(v)-6 b(alue,)37 b(whic)m(h)g(is)f(the)g(area)g(stretc)m(hing)h (factor,)150 2137 y(m)m(ust)28 b(b)s(e)f(the)h(P)m(erron)f(eigen)m(v)-6 b(alue.)43 b(This)27 b(eigen)m(v)-6 b(alue)28 b(is)f(the)h(square)f(of) h(the)f(linear)g(stretc)m(hing)150 2257 y(factor)38 b Fh(c)p Fl(\()p Fh(\036)p Fl(\).\))57 b(Let)38 b Fh(r)i Fl(denote)f(a)e(left)h(eigen)m(v)m(ector)g(of)g Fh(\036)2303 2221 y Fe(\003)2349 2257 y Fl(.)57 b(De\014ne)39 b(the)f(map)g Fh(\026)d Fl(:)g Ff(C)42 b(!)34 b Fg(R)k Fl(b)m(y)150 2376 y(\()p Fh(v)t(;)17 b(k)s Fl(\))26 b Ff(7!)i Fh(\025)592 2340 y Fe(\000)p Fj(k)703 2376 y Fh(r)s(v)t Fl(.)349 2496 y(The)34 b(map)f(is)h(w)m(ell-de\014ned)h(since)1148 2737 y Fh(\026)p Fl(\()p Fh(\036)1306 2696 y Fe(\003)1352 2737 y Fh(v)t(;)17 b(k)24 b Fl(+)f(1\))k(=)h Fh(\025)1904 2696 y Fe(\000)p Fj(k)r Fe(\000)p Fk(1)2118 2737 y Fh(r)s(\036)2225 2696 y Fe(\003)2271 2737 y Fh(v)j Fl(=)d Fh(\025)2513 2696 y Fe(\000)p Fj(k)2625 2737 y Fh(r)s(v)t(:)908 b Fl(5\))150 2979 y(Supp)s(ose)35 b(\()p Fh(v)t(;)17 b(k)s Fl(\))26 b Ff(2)i(C)943 2994 y Fk(+)1032 2979 y Ff(\000)22 b(f)p Fl(0)p Ff(g)32 b Fl(then)i(\()p Fh(\036)1638 2943 y Fe(\003)1684 2979 y Fl(\))1723 2943 y Fj(m)1799 2979 y Fh(v)d Ff(2)d Fl(\()p Fg(Z)2077 2994 y Fk(+)2143 2979 y Fl(\))2182 2943 y Fj(n)2270 2979 y Fl(for)33 b(some)g Fh(m)p Fl(.)45 b(Since)34 b Fh(\036)3150 2943 y Fe(\003)3229 2979 y Fl(is)f(primitiv)m(e,)150 3098 y(w)m(e)g(ma)m(y)g(c)m(ho)s(ose)g Fh(m)h Fl(suc)m(h)g(that)e(all)g(en)m(tries)i(of)f(\()p Fh(\036)2055 3062 y Fe(\003)2101 3098 y Fl(\))2140 3062 y Fj(m)2215 3098 y Fh(v)k Fl(are)c(p)s(ositiv)m(e.)43 b(Since)34 b(all)e(en)m(tries)h(of)g Fh(r)150 3218 y Fl(are)g(non-negativ)m(e,)1103 3340 y Fh(\026)p Fl(\()p Fh(v)t(;)17 b(k)s Fl(\))26 b(=)j Fh(\025)1583 3299 y Fe(\000)p Fj(k)1694 3340 y Fh(r)s(v)i Fl(=)d Fh(\025)1984 3299 y Fe(\000)p Fj(k)r Fe(\000)p Fj(m)2229 3340 y Fh(r)s Fl(\()p Fh(\036)2375 3299 y Fe(\003)2420 3340 y Fl(\))2459 3299 y Fj(m)2535 3340 y Fh(v)j(>)d Fl(0)p Fh(:)864 b Fl(6\))349 3531 y(T)-8 b(o)33 b(pro)m(v)m(e)g(the)h(rev)m(erse)g (direction,)f(tak)m(e)f(a)h(basis)g(of)h Fg(C)2440 3495 y Fj(n)2527 3531 y Fl(whic)m(h)g(includes)g Fh(w)3265 3546 y Fk(0)3310 3531 y Fl(,)f(the)g(righ)m(t)150 3651 y(eigen)m(v)m(ector)e(for)g Fh(\025)p Fl(,)g(and)h(\()p Fh(n)16 b Ff(\000)i Fl(1\))30 b(v)m(ectors)h Ff(f)p Fh(w)1910 3666 y Fk(1)1955 3651 y Fh(;)17 b(w)2071 3666 y Fk(2)2116 3651 y Fh(;)g(:)g(:)g(:)d(;)j(w)2409 3666 y Fj(n)p Fe(\000)p Fk(1)2564 3651 y Ff(g)p Fl(,)31 b(eac)m(h)h Fh(w)2966 3666 y Fj(i)3027 3651 y Ff(2)c Fl(Ker)q(\()p Fh(I)8 b(\025)3431 3666 y Fj(i)3481 3651 y Ff(\000)18 b Fh(\036)3635 3615 y Fe(\003)3681 3651 y Fl(\))3720 3615 y Fj(l)150 3770 y Fl(for)34 b(some)g Fh(l)h Fl(and)f Fh(\025)870 3785 y Fj(i)932 3770 y Ff(6)p Fl(=)29 b Fh(\025)p Fl(.)45 b(W)-8 b(e)34 b(can)g(and)g(do)g(c)m(ho)s(ose)g Fh(w)2244 3785 y Fk(0)2323 3770 y Fl(with)f(all)g(its)g(en)m(tries)i(p)s(ositiv)m (e.)44 b(No)m(w)150 3890 y Fh(r)s(w)269 3905 y Fj(i)330 3890 y Fl(=)28 b(0)33 b(since)1198 4012 y(0)27 b(=)h Fh(r)s Fl(\()p Fh(I)8 b(\025)1577 4027 y Fj(i)1632 4012 y Ff(\000)22 b Fh(\036)1790 3971 y Fe(\003)1837 4012 y Fl(\))1876 3971 y Fj(l)1906 4012 y Fh(w)1977 4027 y Fj(i)2038 4012 y Fl(=)28 b(\()p Fh(\025)2240 4027 y Fj(i)2295 4012 y Ff(\000)23 b Fh(\025)p Fl(\))2492 3971 y Fj(l)2522 4012 y Fh(r)s(w)2641 4027 y Fj(i)2675 4012 y Fh(:)958 b Fl(7\))150 4204 y(Therefore)26 b(if)e Fh(\026)p Fl(\()p Fh(v)t(;)17 b(k)s Fl(\))27 b Fh(>)h Fl(0)c(then)h(writing)f Fh(v)k Fl(in)d(the)f(basis)h Ff(f)p Fh(w)2427 4219 y Fk(0)2472 4204 y Fh(;)17 b(w)2588 4219 y Fk(1)2632 4204 y Fh(;)g(:)g(:)g(:)d(;)j(w)2925 4219 y Fj(n)p Fe(\000)p Fk(1)3081 4204 y Ff(g)24 b Fl(the)h(co)s(e\016cien)m(t)150 4323 y Fh(c)193 4338 y Fk(0)271 4323 y Fl(of)33 b Fh(w)455 4338 y Fk(0)534 4323 y Fl(m)m(ust)g(b)s(e)h(p)s(ositiv)m(e.)43 b(Since)35 b Fh(\025)e Fl(exceeds)h(the)g(mo)s(dulus)g(of)f(all)g (other)g(eigen)m(v)-6 b(alues,)34 b(as)150 4443 y Fh(l)39 b Ff(!)e(1)p Fl(,)j(w)m(e)g(ha)m(v)m(e)g Ff(jj)p Fh(\025)1027 4407 y Fe(\000)p Fj(l)1119 4443 y Fl(\()p Fh(\036)1217 4407 y Fe(\003)1263 4443 y Fl(\))1302 4407 y Fj(l)1332 4443 y Fh(v)30 b Ff(\000)d Fh(c)1557 4458 y Fk(0)1602 4443 y Fh(w)1673 4458 y Fk(0)1718 4443 y Ff(jj)37 b(!)g Fl(0.)61 b(This)40 b(implies)f(\()p Fh(\036)2763 4407 y Fe(\003)2809 4443 y Fl(\))2848 4407 y Fj(l)2878 4443 y Fh(v)j Ff(2)c Fl(\()p Fg(Z)3177 4458 y Fk(+)3243 4443 y Fl(\))3282 4407 y Fj(n)3375 4443 y Fl(for)i(some)150 4562 y Fh(l)r Fl(.)p 269 4572 60 80 v 349 4682 a(Next)j(w)m(e)g(sho)m (w)h(that)f(the)g(map)g Fh(\026)i Fl(:)e Ff(C)51 b(!)44 b Fg(R)f Fl(induces)h(a)f(map)g(from)h Fh(H)51 b Fl(to)42 b Fg(R)p Fl(;)48 b(since)150 4801 y Fh(H)36 b Fl(=)28 b Ff(C)6 b Fh(=)p Fl(Im)33 b Fh(\016)t Fl(,)g(the)h(follo)m(wing)e (will)g(sho)m(w)i(that)f(this)g(map)h(is)f(w)m(ell-de\014ned.)150 4981 y Fm(Lemma)38 b(2)p Fl(.)44 b(Im)17 b Fh(\016)32 b Ff(\022)c Fl(Ker)17 b Fh(\026)p Fl(.)150 5160 y(Pro)s(of:)42 b(Let)30 b Fh(e)g Fl(b)s(e)f(an)h(edge)g(and)g(let)g Fh(f)1580 5175 y Fj(e)1650 5160 y Ff(2)e Fh(C)1822 5124 y Fk(1)1867 5160 y Fl(\(\006)1978 5175 y Fj(x)2028 5160 y Fl(\))h(b)s(e)h(the)g(1-co)s(c)m(hain)g(whic)m(h)g(assigns)g(a)g(v)-6 b(alue)150 5280 y(1)37 b(to)g(the)h(edge)g Fh(e)g Fl(and)f(0)h(to)f (all)f(other)i(edges.)57 b(Then)38 b Fh(\016)t Fl(\()p Fh(f)2389 5295 y Fj(e)2432 5280 y Fl(\))f(is)g(a)g(2-co)s(c)m(hain)h (whic)m(h)g(acts)g(on)1900 5519 y(15)p eop %%Page: 16 17 16 16 bop 150 100 a Fl(the)31 b(tiles)g(as)g(follo)m(ws:)43 b(let)30 b Fh(T)45 b Fl(b)s(e)32 b(an)f Fh(m)p Fl(th)g(lev)m(el)g (tile.)42 b(Then)32 b Fh(\016)t Fl(\()p Fh(f)2567 115 y Fj(e)2610 100 y Fl(\)\()p Fh(T)14 b Fl(\))30 b(coun)m(ts)i(the)f(n)m (um)m(b)s(er)i(of)150 219 y(o)s(ccurences)h(of)e Fh(e)g Fl(on)f(the)h(b)s(oundary)g Fh(T)46 b Fl(with)32 b(a)f Ff(\006)h Fl(signs)g(dep)s(ending)i(up)s(on)e(the)g(orien)m(tation.)150 339 y(In)38 b(particular,)g Ff(j)p Fh(\016)t Fl(\()p Fh(f)936 354 y Fj(e)978 339 y Fl(\)\()p Fh(T)14 b Fl(\))p Ff(j)37 b Fl(is)g(less)h(than)g(or)g(equal)f(to)g(the)h(n)m(um)m(b)s (er)h(of)f(o)s(ccurences)i(of)e Fh(e)g Fl(on)150 458 y(the)c(b)s(oundary)f(of)h(the)f(lev)m(el)g Fh(m)h Fl(tile)e Fh(T)14 b Fl(.)349 578 y(No)m(w)40 b(assume)i Fh(\026)p Fl(\()p Fh(\016)t Fl(\()p Fh(f)1170 593 y Fj(e)1212 578 y Fl(\)\))d Fh(>)g Fl(0.)65 b(Then)41 b(there)g(is)g(an)f Fh(l)i Fl(suc)m(h)g(that)e(for)g(all)g(for)g(an)m(y)g(lev)m(el)150 697 y(0)g(tile)g Fh(T)14 b Fl(,)43 b(\()p Fh(\036)660 661 y Fe(\003)705 697 y Fl(\))744 661 y Fj(l)775 697 y Fh(\016)t Fl(\()p Fh(f)911 712 y Fj(e)953 697 y Fl(\)\()p Fh(T)14 b Fl(\))39 b Fh(>)i Fl(0.)66 b(F)-8 b(urthermore,)43 b(for)e(\014xed)g Fh(T)54 b Fl(this)41 b(quan)m(tit)m(y)f(scales)h(lik) m(e)f Fh(\025)3719 661 y Fj(l)150 817 y Fl(for)k(su\016cien)m(tly)g (large)f Fh(l)r Fl(.)75 b(This)44 b(means)g(that)g(for)g(lev)m(el)f Fh(l)i Fl(tiles)e(there)i(are)e(at)h(least)f Fh(O)s Fl(\()p Fh(\025)3681 781 y Fj(l)3711 817 y Fl(\))150 936 y(o)s(ccurences)35 b(of)e Fh(e)g Fl(on)g(the)g(b)s(oundary)-8 b(.)43 b(Ho)m(w)m(ev)m(er,) 33 b Fh(\025)g Fl(is)g(the)g(area)f(stretc)m(hing)i(factor,)f(so)f (there)150 1056 y(can)25 b(only)g(b)s(e)g Fh(O)s Fl(\()p Fh(\025)838 1020 y Fj(l=)p Fk(2)949 1056 y Fl(\))g(edges)h(on)f(the)g (b)s(oundary)h(of)f(a)g(lev)m(el)g Fh(l)h Fl(tile)f(\(since)h(the)f(p)s (erimeter)g(scales)150 1176 y(lik)m(e)33 b Fh(\025)391 1139 y Fj(l=)p Fk(2)535 1176 y Fl(and)h(there)g(is)f(a)g(minim)m(um)g (length)h(to)f(eac)m(h)h(edge\).)44 b(This)34 b(is)f(a)g(con)m (tradiction.)349 1295 y(A)45 b(similar)g(argumen)m(t)h(sho)m(ws)g(that) f Fh(\026)p Fl(\()p Fh(\016)t Fl(\()p Fh(f)2028 1310 y Fj(e)2070 1295 y Fl(\)\))g(cannot)h(b)s(e)g(negativ)m(e.)80 b(Therefore)47 b(the)150 1415 y(image)33 b(of)g Fh(\016)38 b Fl(is)33 b(a)g(subgroup)h(of)g(the)f(k)m(ernel)h(of)f Fh(\026)h Fl(and)f Ff(C)2279 1430 y Fk(+)2368 1415 y Ff(\\)23 b Fl(Im)16 b Fh(\016)32 b Fl(=)c Ff(f)p Fl(0)p Ff(g)p Fl(.)p 3010 1425 60 80 v 349 1534 a(Abusing)44 b(notation,)g(de\014ne)h Fh(\026)f Fl(:)g Fh(H)51 b Ff(!)44 b Fg(R)f Fl(b)m(y)g Fh(\026)p Fl(\([)p Fh(x)p Fl(]\))g(=)i Fh(\026)p Fl(\()p Fh(x)p Fl(\).)72 b(Letting)43 b Fh(H)3357 1549 y Fk(+)3466 1534 y Fl(b)s(e)h(the)150 1654 y(pre-image)34 b(under)g Fh(\026)f Fl(of)h Fg(R)1171 1669 y Fk(+)1237 1654 y Fl(,)f(\()p Fh(H)s(;)17 b(H)1551 1669 y Fk(+)1615 1654 y Fl(\))33 b(is)g(an)h(ordered)g(group)g(since)172 1773 y(\(1\))49 b Fh(H)432 1788 y Fk(+)520 1773 y Fl(+)23 b Fh(H)703 1788 y Fk(+)797 1773 y Ff(\032)28 b Fh(H)985 1788 y Fk(+)172 1893 y Fl(\(2\))49 b Fh(H)432 1908 y Fk(+)520 1893 y Ff(\\)23 b(\000)p Fh(H)769 1908 y Fk(+)863 1893 y Fl(=)28 b Ff(f)p Fl(0)p Ff(g)172 2012 y Fl(\(3\))49 b Fh(H)432 2027 y Fk(+)520 2012 y Ff(\000)23 b Fh(H)703 2027 y Fk(+)797 2012 y Fl(=)28 b Fh(H)41 b Fl(\(this)33 b(follo)m(ws)g(from)h(the)f(similar)f(prop)s(ert)m(y)i(of)f Ff(C)2893 2027 y Fk(+)2960 2012 y Fl(\).)150 2252 y Fm(4.)51 b(Some)38 b(relations)f(b)s(et)m(w)m(een)e Fh(X)1595 2267 y Fj(x)1645 2252 y Fm(,)j Fh(X)1798 2267 y Fk(0)1880 2252 y Fm(and)g Fh(X)2185 2267 y Fj(\036)2237 2252 y Fm(.)349 2372 y Fl(Before)33 b(w)m(e)f(pro)m(v)m(e)h(the)g(in)m(v)-6 b(ariance)32 b(of)h(P-p)s(ositivit)m(y)d(and)j(extend)g(to)e(arbitrary) h(relativ)m(e)150 2491 y(orien)m(tation)f(groups,)h(w)m(e)f(need)i(to)e (understand)i(the)f(role)f(of)g(rotations)g(in)g(tilings.)42 b(Let)32 b Fh(G)g Fl(b)s(e)150 2611 y(the)i(relativ)m(e)e(orien)m (tation)h(group)g(of)h(a)f(tiling)f(and)i(let)f Fh(R)g Fl(b)s(e)h(the)f(asso)s(ciated)h(group)f(ring.)150 2790 y Fd(Finite)g(r)-5 b(elative)33 b(orientation)f(gr)-5 b(oups.)43 b Fl(W)-8 b(e)31 b(b)s(egin)h(b)m(y)f(assuming)g Fh(G)h Fl(is)f(\014nite.)43 b(As)31 b(previously)150 2910 y(indicated,)48 b Fh(G)p Fl(,)h(and)c(therefore)i Fh(R)q Fl(,)g(acts)e(naturally)g(on)g(the)h(simplicial)e(\(co\)c)m (hain)h(groups)150 3029 y Fh(C)228 2993 y Fj(i)261 3029 y Fl(\(\006)372 3044 y Fj(x)422 3029 y Fl(\))39 b(and)g Fh(C)770 3044 y Fj(i)803 3029 y Fl(\(\006)914 3044 y Fj(x)964 3029 y Fl(\),)h(giving)d(them)i(the)h(structure)f(of)g(mo)s (dules)h(o)m(v)m(er)e Fh(R)q Fl(.)60 b(A)39 b(\014nite)g(index)150 3149 y(subgroup)e(of)e(eac)m(h)h(mo)s(dule)g(can)f(b)s(e)h(written)f (as)g(a)g(direct)h(sum)g(of)f(irreducible)h(represen)m(ta-)150 3268 y(tions)41 b(of)g Fh(G)p Fl(.)69 b(The)41 b(substitution)h(maps)f (are)h(equiv)-6 b(arian)m(t)40 b(and)i(so)f(are)h(mo)s(dule)f(homomor-) 150 3388 y(phisms,)35 b(mapping)f(eac)m(h)g(irreducible)h(represen)m (tation)g(to)f(itself.)45 b(The)35 b(\(co\)b)s(oundary)f(maps)150 3508 y(are)27 b(also)g(equiv)-6 b(arian)m(t,)28 b(and)g(so)f(map)h(a)f (represen)m(tation)h(in)f Fh(C)2475 3471 y Fj(i)2536 3508 y Fl(\(or)g Fh(C)2762 3523 y Fj(i)2795 3508 y Fl(\))g(to)g(the)h (corresp)s(onding)150 3627 y(represen)m(tation)42 b(in)g Fh(C)1012 3591 y Fj(i)p Fk(+1)1187 3627 y Fl(\(or)f Fh(C)1427 3642 y Fj(i)p Fe(\000)p Fk(1)1563 3627 y Fl(\).)67 b(Th)m(us)43 b(the)e(in)m(teger)h(\(co\)homology)e(groups,)j(and)f(in)150 3747 y(particular)d(the)g(top)g(cohomology)-8 b(,)39 b(are)g(themselv)m(es)h(\014nite)g(extensions)f(of)h(\014nite)f (quotien)m(ts)150 3866 y(of)33 b(direct)h(sums)g(of)g(irreducible)g (represen)m(tations.)349 3986 y(The)50 b(need)g(for)f(\014nite)h (extensions)f(and)h(\014nite)f(quotien)m(ts)g(comes)h(ab)s(out)e(as)h (follo)m(ws.)150 4105 y(When)40 b(a)f(compact)h(group)f(acts)g(on)g Fg(R)1655 4069 y Fj(n)1710 4105 y Fl(,)h Fg(R)1850 4069 y Fj(n)1943 4105 y Fl(splits)f(up)h(as)f(the)h(direct)f(sum)h(of)f (irreducible)150 4225 y(represen)m(tations)29 b(of)e(that)g(group.)42 b(Ho)m(w)m(ev)m(er,)29 b(the)e(same)h(observ)-6 b(ation)27 b(do)s(es)h(not)f(apply)g(to)g Fg(Z)3668 4189 y Fj(n)3722 4225 y Fl(.)150 4344 y(F)-8 b(or)41 b(example,)i(if)f Fg(Z)931 4359 y Fk(2)1017 4344 y Fl(acts)g(on)f Fg(Z)1443 4308 y Fk(2)1530 4344 y Fl(b)m(y)h(p)s(erm)m(uting)g(the)f(t)m(w)m(o)h (co)s(ordinates,)h(the)f(irreducible)150 4464 y(represen)m(tations)g (are)f(all)f(m)m(ultiples)h(of)h(\(1)p Fh(;)17 b Fl(1\))39 b(and)i(all)f(m)m(ultiples)h(of)g(\(1)p Fh(;)17 b Ff(\000)p Fl(1\).)66 b(The)42 b(direct)150 4583 y(sum)j(of)g(these)g(t)m(w)m(o)f (represen)m(tations)i(giv)m(es)e(all)g(elemen)m(ts)h(\()p Fh(a;)17 b(b)p Fl(\))43 b(with)h Fh(a)29 b Fl(+)h Fh(b)44 b Fl(ev)m(en)h(|)f(an)150 4703 y(index)31 b(2)g(subgroup)i(of)f Fg(Z)1106 4667 y Fk(2)1151 4703 y Fl(.)43 b(Similarly)-8 b(,)29 b(\014nite)j(index)f(subgroups)i(of)f(the)f(k)m(ernels)h(and)g (images)150 4823 y(of)47 b(the)h(\(co\)b)s(oundary)f(maps)g(are)g (direct)h(sums)g(of)f(represen)m(tations)i(of)e Fh(G)p Fl(.)86 b(Computing)150 4942 y(\(co\)homology)43 b(one)i(represen)m (tation)h(at)e(a)g(time)h(is)f(equiv)-6 b(alen)m(t)45 b(to)f(taking)f(a)i(\014nite)g(index)150 5062 y(subgroup)i(of)e(the)h (k)m(ernel)g(of)f Fh(\016)50 b Fl(and)c(mo)s(dding)f(out)g(b)m(y)h(a)f (\014nite)h(index)f(subgroup)i(of)f(the)150 5181 y(image)37 b(of)g Fh(\016)t Fl(.)57 b(Correcting)37 b(for)g(our)h(taking)e(to)s(o) g(small)h(a)g(n)m(umerator)h(means)g(doing)g(a)f(\014nite)1900 5519 y(16)p eop %%Page: 17 18 17 17 bop 150 100 a Fl(extension.)44 b(Correcting)33 b(for)h(the)g(denominator)f(in)m(v)m(olv)m(es)g(taking)f(a)h(\014nite)h (quotien)m(t.)349 219 y(Since)28 b(the)e(substitution)h(map)f(is)h (equiv)-6 b(arian)m(t,)27 b(rotating)e(an)i(eigen)m(v)m(ector)f(giv)m (es)h(another)150 339 y(eigen)m(v)m(ector)c(with)g(the)g(same)g(eigen)m (v)-6 b(alue.)41 b(Since)24 b(the)f(P)m(erron)g(eigen)m(v)m(ectors)g (are)g(unique,)i(with)150 458 y(purely)45 b(p)s(ositiv)m(e)f(en)m (tries,)j(they)e(m)m(ust)g(b)s(e)g(rotationally)d(in)m(v)-6 b(arian)m(t)44 b(and)h(therefore)h(b)s(elong)150 578 y(to)d(the)i(trivial)d(represen)m(tation.)77 b(Since)44 b(the)g(left)g(P)m(erron)h(eigen)m(v)m(ector)f(is)g(in)m(v)-6 b(arian)m(t,)46 b Fh(\026)e Fl(of)150 697 y(a)g(co)s(c)m(hain)i(is)e (also)g(rotationally)f(in)m(v)-6 b(arian)m(t.)78 b(But)44 b(that)g(implies)h(that)f Fh(\026)h Fl(acting)f(on)h(eac)m(h)150 817 y(non)m(trivial)32 b(represen)m(tation)j(is)e(iden)m(tically)f (zero.)349 936 y(F)-8 b(or)29 b(example)g(in)g(the)g(uncollared)h(c)m (hair)f(complex)g(there)h(is)f(just)g(one)h(tile,)f(app)s(earing)g(in) 150 1056 y(four)35 b(di\013eren)m(t)h(orien)m(tations,)e(and)h Fh(C)1602 1020 y Fk(2)1677 1056 y Ff(\030)30 b Fh(R)g Fl(=)g Fg(Z)p Fl([)p Fh(t)p Fl(])p Fh(=)p Fl(\()p Fh(t)2280 1020 y Fk(4)2347 1056 y Ff(\000)24 b Fl(1\))34 b(where)h Fh(t)f Fl(represen)m(ts)j(rotation)150 1176 y(b)m(y)32 b Fh(\031)t(=)p Fl(2.)42 b(The)32 b(substitution)g(map)g(is)f(giv)m(en) h(b)m(y)g(the)f(1)19 b Ff(\002)g Fl(1)32 b(matrix)e(2)19 b(+)g Fh(t)g Fl(+)g Fh(t)3078 1139 y Fk(3)3123 1176 y Fl(.)43 b(In)32 b(the)g(trivial)150 1295 y(represen)m(tation)f(w)m(e)g (set)g Fh(t)c Fl(=)h(1,)j(so)f(our)g(matrix)f(has)i(P)m(erron)g(eigen)m (v)-6 b(alue)31 b(4,)f(with)g(the)g(P)m(erron)150 1415 y(eigen)m(v)m(ector)f(b)s(eing)f(an)m(y)h(m)m(ultiple)f(of)h(an)f(in)m (v)-6 b(arian)m(t)28 b(elemen)m(t)i(of)e Fh(C)2700 1379 y Fk(2)2745 1415 y Fl(,)h Fd(i.e.)41 b Fl(of)29 b(\(1)12 b(+)g Fh(t)g Fl(+)g Fh(t)3454 1379 y Fk(2)3513 1415 y Fl(+)g Fh(t)3638 1379 y Fk(3)3684 1415 y Fl(\).)349 1534 y(This)46 b(suggests)g(a)f(streamlined)h(computation)f(of)g(the)h(P)m (erron)g(eigen)m(v)-6 b(alue)46 b(and)f(P)m(er-)150 1654 y(ron)40 b(eigen)m(v)m(ector.)63 b(Instead)41 b(of)e(w)m(orking)g(on)h (all)f(of)h(\006)2252 1669 y Fj(x)2302 1654 y Fl(,)h(w)m(ork)e(on)g (the)h(simplicial)f(complex)150 1773 y(\006)222 1788 y Fk(0)302 1773 y Fl(=)d(\006)487 1788 y Fj(x)537 1773 y Fh(=G)p Fl(,)j(whose)g(\(co\)c)m(hain)f(groups)h(are)f(precisely)g (the)h(trivial)d(represen)m(tations)j(of)g Fh(G)150 1893 y Fl(within)30 b(the)h(\(co\)c)m(hain)f(groups)h(of)g(\006)1559 1908 y Fj(x)1609 1893 y Fl(.)43 b Fh(C)1758 1857 y Fk(2)1803 1893 y Fl(\(\006)1914 1908 y Fj(x)1964 1893 y Fl(\))16 b Ff(\012)2096 1908 y Fb(Z)2164 1893 y Fg(R)30 b Fl(equals)g Fh(C)2643 1857 y Fk(2)2688 1893 y Fl(\(\006)2799 1908 y Fk(0)2844 1893 y Fl(\))16 b Ff(\012)2976 1908 y Fb(Z)3044 1893 y Fg(R)30 b Fl(plus)h(the)g(other)150 2012 y(represen)m(tations)42 b(that)d(app)s(ear)i(in)f Fh(C)1608 1976 y Fk(2)1653 2012 y Fl(\(\006)1764 2027 y Fj(x)1814 2012 y Fl(\).)64 b(The)41 b(P)m(erron)f(eigen)m(v)-6 b(alue)41 b(on)f Fh(C)3195 1976 y Fk(2)3240 2012 y Fl(\(\006)3351 2027 y Fj(x)3401 2012 y Fl(\))27 b Ff(\012)g Fg(R)40 b Fl(is)150 2132 y(precisely)47 b(the)f(P)m(erron)h(eigen)m(v)-6 b(alue)47 b(on)g Fh(C)1813 2096 y Fk(2)1858 2132 y Fl(\(\006)1969 2147 y Fk(0)2013 2132 y Fl(\))31 b Ff(\012)g Fg(R)p Fl(,)49 b(and)e(the)g(\(left)f(and)h(righ)m(t\))e(P)m(erron)150 2252 y(eigen)m(v)m(ectors)34 b(on)f Fh(C)923 2215 y Fk(2)968 2252 y Fl(\(\006)1079 2267 y Fj(x)1129 2252 y Fl(\))22 b Ff(\012)h Fg(R)33 b Fl(come)h(from)f(P)m(erron)h(eigen)m(v)m(ectors)g (on)f Fh(C)2984 2215 y Fk(2)3029 2252 y Fl(\(\006)3140 2267 y Fk(0)3185 2252 y Fl(\))22 b Ff(\012)g Fg(R)p Fl(.)349 2371 y(Since)39 b(the)f(\(co\)c)m(hain)g(groups)h(of)f(\006)1746 2386 y Fk(0)1828 2371 y Fl(are)g(isomorphic)g(to)g(the)g(in)m(v)-6 b(arian)m(t)38 b(parts)g(of)g(the)150 2491 y(\(co\)c)m(hain)49 b(groups)h(of)f(\006)1139 2506 y Fj(x)1189 2491 y Fl(,)j(and)d(since)h (the)f(\(co\)b)s(oundary)g(and)g(substitution)h(maps)f(are)150 2610 y(equiv)-6 b(arian)m(t,)32 b(w)m(e)i(ha)m(v)m(e)150 2790 y Fm(Theorem)46 b(2.)68 b Fd(Up)43 b(to)g(\014nite)f(extensions,)h (the)g(\(c)-5 b(o\)homolo)g(gy)42 b(of)g Fl(\006)2866 2805 y Fk(0)2954 2790 y Fd(is)g(e)-5 b(qual)43 b(to)g(the)g(r)-5 b(o-)150 2909 y(tational)5 b(ly)43 b(invariant)d(p)-5 b(art)42 b(of)f(the)h(\(c)-5 b(o\)homolo)g(gy)42 b(of)f Fl(\006)2358 2924 y Fj(x)2408 2909 y Fd(,)i(and)f(the)f(c)-5 b(ohomolo)g(gy)42 b(of)g Fh(X)3593 2924 y Fk(0)3679 2909 y Fd(is)150 3029 y(isomorphic)34 b(to)i(the)g(r)-5 b(otational)5 b(ly)36 b(invariant)f(p)-5 b(art)35 b(of)g(the)h(c)-5 b(ohomolo)g(gy)36 b(of)f Fh(X)3128 3044 y Fj(x)3177 3029 y Fd(.)349 3208 y Fl(W)-8 b(e)35 b(cannot)g(mak)m(e)g(an)m(y)f (statemen)m(ts)i(ab)s(out)e(the)h(homologies)f(of)h Fh(X)2973 3223 y Fk(0)3052 3208 y Fl(and)g Fh(X)3330 3223 y Fj(x)3415 3208 y Fl(b)s(ecause)150 3328 y(homology)-8 b(,)31 b(unlik)m(e)j (cohomology)-8 b(,)31 b(do)s(es)j(not)f(b)s(eha)m(v)m(e)h(w)m(ell)f (under)i(in)m(v)m(erse)f(limits.)349 3448 y(No)m(w)46 b Fh(X)672 3463 y Fj(\036)770 3448 y Fl(is)g(a)f(circle)h(\014bration)g (o)m(v)m(er)g Fh(X)1977 3463 y Fk(0)2021 3448 y Fl(,)j(with)c(singular) h(\014b)s(ers)h(corresp)s(onding)f(to)150 3567 y(tilings)33 b(that)f(are)i(symmetric)f(ab)s(out)g(the)h(origin.)43 b(\(These)34 b(singular)g(\014b)s(ers)g(ha)m(v)m(e)g(\014nite)g(m)m (ul-)150 3687 y(tiplicit)m(y)-8 b(,)34 b(since)i(an)m(y)g(giv)m(en)f (tiling)f(can)i(only)f(admit)g(discrete)h(rotational)e(symmetry)h(ab)s (out)150 3806 y(the)k(origin.\))59 b(The)39 b(existence)g(of)g(these)g (singular)g(\014b)s(ers)h(means)f(that)f(computing)h Fh(\031)3465 3821 y Fk(1)3509 3806 y Fl(\()p Fh(X)3631 3821 y Fj(\036)3684 3806 y Fl(\),)150 3926 y Fh(H)241 3890 y Fk(1)286 3926 y Fl(\()p Fh(X)408 3941 y Fj(\036)460 3926 y Fl(\))26 b(and)i Fh(H)804 3890 y Fk(2)848 3926 y Fl(\()p Fh(X)970 3941 y Fj(\036)1023 3926 y Fl(\))e(from)h(corresp)s (onding)h(data)f(on)g Fh(X)2387 3941 y Fk(0)2458 3926 y Fl(can)h(b)s(e)f(complicated.)42 b(Ho)m(w)m(ev)m(er,)150 4045 y(computing)27 b(the)h(top)f(cohomology)e Fh(H)1602 4009 y Fk(3)1647 4045 y Fl(\()p Fh(X)1769 4060 y Fj(\036)1821 4045 y Fl(\))i(is)g(quite)f(easy)-8 b(.)42 b(F)-8 b(rom)27 b(the)g(sp)s(ectral)g(sequence)i(of)150 4165 y(the)36 b(\014bration,)g(it)g(follo)m(ws)f(immediately)g(that)h Fh(H)2071 4129 y Fk(3)2115 4165 y Fl(\()p Fh(X)2237 4180 y Fj(\036)2290 4165 y Fl(\))f(is)h(isomorphic)g(to)f Fh(H)3189 4129 y Fk(2)3234 4165 y Fl(\()p Fh(X)3356 4180 y Fk(0)3400 4165 y Fl(\))h([BoT].)150 4284 y(F)-8 b(rom)33 b(this,)g(and)g(from)h(the)g(fact)f(that)g(\006)1719 4299 y Fj(\036)1805 4284 y Fl(is)g(a)g(circle)h(\014bration)f(o)m(v)m (er)g(\006)2938 4299 y Fk(0)2983 4284 y Fl(,)g(w)m(e)g(ha)m(v)m(e)150 4464 y Fm(Corollary)51 b(1)p Fl(.)76 b Fd(Up)45 b(to)g(\014nite)g (extensions,)h(the)g(top)f(c)-5 b(ohomolo)g(gy)45 b(of)g Fl(\006)3014 4479 y Fj(\036)3112 4464 y Fd(is)g(e)-5 b(qual)46 b(to)f(the)150 4583 y(r)-5 b(otational)5 b(ly)45 b(invariant)e(p)-5 b(art)44 b(of)f(the)h(top)g(c)-5 b(ohomolo)g(gy)45 b(of)e Fl(\006)2559 4598 y Fj(x)2609 4583 y Fd(.)71 b(Up)44 b(to)g(\014nite)g(extensions,)150 4703 y(the)32 b(top)h(c)-5 b(ohomolo)g(gy)33 b(of)f Fh(X)1201 4718 y Fj(\036)1286 4703 y Fd(is)f(isomorphic)h(to)g(the)h(r)-5 b(otational)5 b(ly)33 b(invariant)e(p)-5 b(art)33 b(of)f(the)g(top)150 4823 y(c)-5 b(ohomolo)g(gy)36 b(of)f Fh(X)877 4838 y Fj(x)927 4823 y Fd(.)349 5002 y Fl(Note)k(also)g(that)g(the)h (top-dimensional)g(substitution)g(matrix)e(for)h(\006)3055 5017 y Fj(\036)3148 5002 y Fl(is)g(iden)m(tical)g(to)150 5122 y(the)g(top-dimensional)g(substitution)g(matrix)e(for)i(\006)2177 5137 y Fk(0)2260 5122 y Fl(|)f(the)h Fh(S)2642 5086 y Fk(1)2725 5122 y Fl(factor)g(just)g(comes)h(along)150 5241 y(for)h(the)g(ride.)66 b(W)-8 b(e)41 b(ma)m(y)f(therefore)i (compute)f(our)g(order)g(structure)g(on)g(whic)m(hev)m(er)h(of)f(the) 1900 5519 y(17)p eop %%Page: 18 19 18 18 bop 150 100 a Fl(three)34 b(spaces)g(\006)782 115 y Fj(x)833 100 y Fl(,)e(\006)965 115 y Fk(0)1043 100 y Fl(or)h(\006)1237 115 y Fj(\036)1323 100 y Fl(is)h(most)f(con)m(v)m (enien)m(t.)349 219 y(T)-8 b(o)22 b(illustrate)g(these)h(principles,)j (w)m(e)c(compute)i(the)f(cohomology)e(of)i(the)g(P)m(enrose)g(tilings.) 150 339 y(Here)47 b(the)h(relativ)m(e)f(orien)m(tation)f(group)i(is)f Fg(Z)1951 354 y Fk(10)2036 339 y Fl(,)j(with)d(group)h(ring)f Fh(R)k Fl(=)h Fg(Z)p Fl([)p Fh(t)p Fl(])p Fh(=)p Fl(\()p Fh(t)3410 303 y Fk(10)3525 339 y Ff(\000)32 b Fl(1\),)150 458 y(where)g Fh(t)g Fl(represen)m(ts)h(rotation)e(b)m(y)h(2)p Fh(\031)t(=)p Fl(10.)42 b(There)32 b(are)g(4)f(irreducible)h(represen)m (tations)h(of)f Fh(G)p Fl(,)150 578 y(corresp)s(onding)i(to)e(the)h (factorization)f Fh(t)1696 542 y Fk(10)1802 578 y Ff(\000)21 b Fl(1)28 b(=)g(\()p Fh(t)21 b Ff(\000)g Fl(1\)\()p Fh(t)f Fl(+)i(1\)\()p Fh(t)2724 542 y Fk(4)2789 578 y Fl(+)f Fh(t)2923 542 y Fk(3)2989 578 y Fl(+)h Fh(t)3124 542 y Fk(2)3189 578 y Fl(+)g Fh(t)f Fl(+)g(1\)\()p Fh(t)3607 542 y Fk(4)3673 578 y Ff(\000)150 697 y Fh(t)186 661 y Fk(3)254 697 y Fl(+)j Fh(t)391 661 y Fk(2)459 697 y Ff(\000)f Fh(t)g Fl(+)h(1\).)49 b(The)35 b(\014rst)g(t)m(w)m(o)g(are)g (one)h(dimensional)f(and)g Fh(t)g Fl(acts)g(b)m(y)g(m)m(ultiplication)f (b)m(y)150 817 y Ff(\006)p Fl(1.)47 b(The)35 b(other)f(t)m(w)m(o)h (represen)m(tations)g(are)g(4)f(dimensional)g(and)h(corresp)s(ond)h (algebraically)150 936 y(to)d Fg(Z)p Fl([)p Fh(t)p Fl(])p Fh(=)p Fl(\()p Fh(t)555 900 y Fk(4)621 936 y Fl(+)22 b Fh(t)756 900 y Fk(3)823 936 y Fl(+)h Fh(t)959 900 y Fk(2)1026 936 y Fl(+)f Fh(t)g Fl(+)h(1\))33 b(and)h Fg(Z)p Fl([)p Fh(t)p Fl(])p Fh(=)p Fl(\()p Fh(t)1882 900 y Fk(4)1948 936 y Ff(\000)22 b Fh(t)2083 900 y Fk(3)2150 936 y Fl(+)h Fh(t)2286 900 y Fk(2)2353 936 y Ff(\000)f Fh(t)g Fl(+)h(1\).)349 1056 y(In)36 b(this)f(tiling,)f(there)i(are)f(4)g(t)m(yp)s(es)g(of)g (tiles,)g(whose)h(geometry)f(and)g(substitutions)h(are)150 1176 y(sho)m(wn)27 b(in)g(Fig.)e(9.)42 b(Similarly)-8 b(,)25 b(there)i(are)g(four)g(kinds)g(of)f(edges,)j(eac)m(h)e(in)f(10)g (orien)m(tations,)h(and)150 1295 y(four)36 b(kinds)g(of)g(v)m(ertices,) h(whic)m(h)f(w)m(e)g(lab)s(el)g Fh(\013;)17 b(\014)5 b(;)17 b(\015)6 b(;)17 b(\016)t Fl(,)34 b(with)h(the)h(relations)f Fh(\013)d Fl(=)h Fh(t\014)5 b Fl(,)36 b Fh(\014)i Fl(=)32 b Fh(t\013)p Fl(,)150 1415 y Fh(\015)52 b Fl(=)47 b Fh(t\016)h Fl(and)d Fh(\016)51 b Fl(=)c Fh(t\015)6 b Fl(.)77 b(The)45 b(c)m(hain)g(groups)g(of)g(\006)2149 1430 y Fj(x)2244 1415 y Fl(are)f(therefore)i Fh(C)2922 1430 y Fk(2)3013 1415 y Fl(=)h Fh(C)3208 1430 y Fk(1)3300 1415 y Fl(=)g Fh(R)3501 1379 y Fk(4)3589 1415 y Fl(and)150 1534 y Fh(C)221 1549 y Fk(0)294 1534 y Fl(=)28 b(\()p Fg(Z)p Fl([)p Fh(t)p Fl(])p Fh(=)p Fl(\()p Fh(t)721 1498 y Fk(2)787 1534 y Ff(\000)22 b Fl(1\)\))1014 1498 y Fk(2)1058 1534 y Fl(.)44 b(The)34 b(b)s(oundary)g(maps)f(are)h(giv)m(en)f(b)m(y)g(the)h (matrices)508 1868 y Fh(@)561 1883 y Fk(1)634 1868 y Fl(=)739 1728 y Fi(\022)829 1808 y Fl(1)22 b Ff(\000)g Fh(t)100 b Ff(\000)p Fl(1)g Ff(\000)p Fh(t)g Ff(\000)p Fl(1)908 1928 y(0)217 b(1)170 b(1)177 b Fh(t)1720 1728 y Fi(\023)1810 1868 y Fl(;)215 b Fh(@)2106 1883 y Fk(2)2179 1868 y Fl(=)2284 1638 y Fi(0)2284 1813 y(B)2284 1877 y(@)2388 1689 y Ff(\000)p Fl(1)161 b Fh(t)177 b(t)2925 1653 y Fk(4)3085 1689 y Ff(\000)p Fh(t)3198 1653 y Fk(7)2388 1808 y Ff(\000)p Fl(1)138 b Fh(t)2689 1772 y Fk(9)2872 1808 y Ff(\000)p Fh(t)h(t)3160 1772 y Fk(8)2426 1928 y Fl(1)f Ff(\000)p Fh(t)2727 1892 y Fk(5)2904 1928 y Fl(0)186 b(0)2426 2047 y(0)193 b(0)185 b(1)131 b Ff(\000)p Fh(t)3198 2011 y Fk(5)3260 1638 y Fi(1)3260 1813 y(C)3260 1877 y(A)3364 1868 y Fh(;)269 b Fl(8\))150 2202 y(where)32 b(w)m(e)f(ha)m(v)m(e)g(tak)m(en)g(\()p Fh(\013;)17 b(\015)6 b Fl(\))28 b(as)j(our)g(basis)g(for)g Fh(C)2107 2217 y Fk(0)2152 2202 y Fl(.)43 b(The)31 b(cob)s(oundary)g(maps)g(are)g(giv) m(en)g(b)m(y)150 2321 y(the)k(transp)s(oses)i(of)e(these)h(matrices,)f (with)g Fh(t)g Fl(replaced)i(throughout)e(b)m(y)g Fh(t)3007 2285 y Fe(\000)p Fk(1)3114 2321 y Fl(.)50 b(The)36 b(ranks)f(of)150 2441 y(these)e(matrices)f(are)g(easily)f(computed)i(b)m(y)f(ro)m(w)g (reduction,)h(one)f(represen)m(tation)h(at)f(a)g(time:)172 2561 y(\(1\))49 b(If)30 b Fh(t)e Fl(=)g(1,)i(then)h Fh(@)1005 2524 y Fe(\003)999 2585 y Fk(2)1080 2561 y Fl(has)f(rank)f(2)h(and)g Fh(@)1809 2524 y Fe(\003)1803 2585 y Fk(1)1884 2561 y Fl(has)g(rank)g(1,)g(so)g Fh(H)2603 2524 y Fk(0)2647 2561 y Fl(\(\006)2758 2576 y Fj(x)2808 2561 y Fl(\))g(and)g Fh(H)3158 2524 y Fk(1)3202 2561 y Fl(\(\006)3313 2576 y Fj(x)3363 2561 y Fl(\))g(con)m(tain)349 2680 y(one)k(cop)m(y)f(of)h (this)f(represen)m(tation)i(while)e Fh(H)2075 2644 y Fk(2)2119 2680 y Fl(\(\006)2230 2695 y Fj(x)2280 2680 y Fl(\))g(con)m(tains)h(t)m(w)m(o.)172 2800 y(\(2\))49 b(If)29 b Fh(t)f Fl(=)g Ff(\000)p Fl(1,)i(then)f Fh(@)1079 2764 y Fe(\003)1073 2824 y Fk(2)1153 2800 y Fl(and)h Fh(@)1402 2764 y Fe(\003)1396 2824 y Fk(1)1476 2800 y Fl(ha)m(v)m(e)f(rank)f(t)m(w)m(o,)i(so)e Fh(H)2347 2764 y Fk(2)2392 2800 y Fl(\(\006)2503 2815 y Fj(x)2553 2800 y Fl(\))g(con)m(tains)i(t)m(w)m(o)e(copies)i(of)f(this)349 2919 y(represen)m(tation)35 b(while)e Fh(H)1352 2883 y Fk(1)1396 2919 y Fl(\(\006)1507 2934 y Fj(x)1557 2919 y Fl(\))g(and)h Fh(H)1914 2883 y Fk(0)1959 2919 y Fl(\(\006)2070 2934 y Fj(x)2119 2919 y Fl(\))f(con)m(tain)h(none.)172 3039 y(\(3\))49 b(If)34 b Fh(t)485 3003 y Fk(4)551 3039 y Fl(+)22 b Fh(t)686 3003 y Fk(3)752 3039 y Fl(+)h Fh(t)888 3003 y Fk(2)954 3039 y Fl(+)f Fh(t)g Fl(+)g(1)27 b(=)h(0,)33 b(then)h Fh(@)1789 3003 y Fe(\003)1783 3064 y Fk(1)1867 3039 y Fl(is)f(zero,)g(since)h Fh(C)2520 3054 y Fk(0)2597 3039 y Fl(\(also)f Fh(C)2914 3003 y Fk(0)2959 3039 y Fl(\))f(do)s(es)i(not)f(con)m(tain)349 3158 y(this)h(represen)m (tation.)45 b Fh(@)1293 3122 y Fe(\003)1287 3183 y Fk(2)1371 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b(The)41 b(nerv)m(e)150 1056 y(of)h(an)g(op)s(en)h(co)m(v)m(er)f(is)g(constructed)h(b)m(y)f (including)g(one)h(v)m(ertex)e(for)h(eac)m(h)h(op)s(en)g(set)f(in)g (the)150 1176 y(co)m(v)m(er,)32 b(then)h(including)f(a)g Fh(k)s Fl(-simplex)g(b)s(et)m(w)m(een)i(an)m(y)d Fh(k)s Fl(-v)m(ertices)i(suc)m(h)g(that)f(the)g(in)m(tersection)150 1295 y(of)c(the)h(corresp)s(onding)g Fh(k)j Fl(sets)c(in)h(the)f(op)s (en)h(co)m(v)m(er)g(is)f(nonempt)m(y)-8 b(.)42 b(The)2899 1270 y(\024)2888 1295 y(Cec)m(h)30 b(cohomology)d(of)150 1415 y(the)36 b(co)m(v)m(er)g(is)g(de\014ned)i(as)e(the)g(simplicial)f (cohomology)g(of)h(the)g(nerv)m(e)g(of)g(that)g(co)m(v)m(er.)52 b(Since)150 1534 y(ev)m(ery)35 b(v)m(ertex)f(of)i(the)f(triangulation)f (lies)g(in)h(exactly)f(one)h(op)s(en)h(star)f(and)g(ev)m(ery)g Fh(k)s Fl(-simplex)150 1654 y(lies)j(in)h(exactly)e Fh(k)k Fl(op)s(en)e(stars,)h(the)e(nerv)m(e)h(of)g(the)g(op)s(en)g(star)f(co)m (v)m(er)h(of)f(\006)3057 1669 y Fj(n)3150 1654 y Fl(is)g(iden)m(tical)g (to)150 1773 y(the)c(triangulation)f(of)i(\006)1108 1788 y Fj(n)1196 1773 y Fl(as)f(a)g(simplicial)f(complex.)47 b(Th)m(us)35 b(the)2715 1748 y(\024)2704 1773 y(Cec)m(h)g(cohomology)e (of)i(the)150 1893 y(op)s(en)29 b(star)g(co)m(v)m(er)g(of)g(\006)1016 1908 y Fj(n)1099 1893 y Fl(is)f Fd(c)-5 b(anonic)g(al)5 b(ly)29 b Fl(iden)m(ti\014ed)h(with)e(the)h(simplicial)f(cohomology)g (of)h(the)150 2012 y(triangulation)39 b(of)h(\006)947 2027 y Fj(n)1001 2012 y Fl(.)64 b(\(This)40 b(argumen)m(t)h(sho)m(ws)g (that)e(simplicial)h(and)2996 1987 y(\024)2985 2012 y(Cec)m(h)h (cohomology)150 2132 y(agree)33 b(for)h(all)e(simplicial)h (complexes.\))349 2252 y(Consider)24 b(triangulations)f(of)g(\006)1558 2267 y Fj(n)1612 2252 y Fl(,)i(generated)f(recursiv)m(ely)-8 b(,)25 b(so)f(that)f(eac)m(h)h(triangle)e(in)i(the)150 2371 y(triangulation)g(of)h(\006)917 2386 y Fj(n)997 2371 y Fl(maps)g(b)m(y)h Fh(\036)f Fl(to)g(a)g(single)g(triangle)g(in)g (the)g(\014rst)h(barycen)m(tric)g(sub)s(division)150 2491 y(of)i(the)g(triangulation)f(of)h(\006)1197 2506 y Fj(n)p Fe(\000)p Fk(1)1353 2491 y Fl(.)42 b(This)28 b(triangulation)f(of)h(\006)2418 2506 y Fj(n)2500 2491 y Fl(corresp)s(onds)h(to)e(an)h(op)s(en)h(co)m(v)m(er)150 2610 y(of)38 b(\006)340 2625 y Fj(n)395 2610 y Fl(,)h(whic)m(h)f(pulls) h(bac)m(k)f(b)m(y)h(the)f(natural)g(pro)6 b(jection)38 b Fh(\031)2441 2625 y Fj(n)2531 2610 y Fl(:)e Fh(X)2678 2625 y Fj(x)2763 2610 y Ff(!)g Fl(\006)2971 2625 y Fj(n)3063 2610 y Fl(to)i(giv)m(e)f(an)h(op)s(en)150 2730 y(co)m(v)m(er)24 b Ff(A)478 2745 y Fj(n)556 2730 y Fl(of)g Fh(X)743 2745 y Fj(x)793 2730 y Fl(.)41 b(That)23 b(is,)j(an)e(op)s(en)g(set)h(in)f Ff(A)1916 2745 y Fj(n)1993 2730 y Fl(is)g(the)g(set)h(of)f(tilings)f (in)h(whic)m(h)g(the)h(origin)e(sits)150 2849 y(in)32 b(a)f(sp)s(eci\014ed)i(neigh)m(b)s(orho)s(o)s(d)f(in)g(a)f(tile)g(of)h (lev)m(el)f Fh(n)p Fl(.)44 b(These)32 b(neigh)m(b)s(orho)s(o)s(ds,)h (as)e(de\014ned)j(b)m(y)150 2969 y(the)f(triangulations)g(of)g(\006) 1144 2984 y Fj(n)1198 2969 y Fl(,)g(are)g(increasingly)g(\014ne)h(as)f Fh(n)28 b Ff(!)f(1)33 b Fl(\(ev)m(en)h(after)f(accoun)m(ting)h(for)150 3088 y(the)f(expansion)f(due)h(to)f Fh(\036)p Fl(\),)g(so)h(the)f (sequence)i(of)e(op)s(en)h(co)m(v)m(ers)g Ff(A)2680 3103 y Fj(n)2766 3088 y Fl(generates)g(the)g(top)s(ology)150 3208 y(of)g Fh(X)346 3223 y Fj(x)396 3208 y Fl(.)349 3328 y(The)574 3302 y(\024)563 3328 y(Cec)m(h)43 b(cohomology)f(of)g Fh(X)1578 3343 y Fj(x)1670 3328 y Fl(is)g(the)h(direct)f(limit)f(of)i (the)2813 3302 y(\024)2802 3328 y(Cec)m(h)g(cohomologies)f(of)150 3447 y(the)36 b(co)m(v)m(ers)h Ff(A)704 3462 y Fj(n)793 3447 y Fl(\(hence)g(the)g(direct)f(limit)e(of)i(the)h(simplicial)d (cohomologies)i(of)g(\006)3342 3462 y Fj(n)3396 3447 y Fl(\))g(with)f(a)150 3567 y(b)s(onding)j(map)g(de\014ned)h(as)f (follo)m(ws.)56 b(F)-8 b(rom)37 b(their)h(construction,)h Ff(A)2810 3582 y Fj(n)p Fk(+1)3002 3567 y Fl(re\014nes)g Ff(A)3397 3582 y Fj(n)3451 3567 y Fl(.)57 b(Eac)m(h)150 3686 y(elemen)m(t)39 b(of)g Ff(A)722 3701 y Fj(n)815 3686 y Fl(\(resp.)61 b Ff(A)1200 3701 y Fj(n)p Fk(+1)1354 3686 y Fl(\))39 b(is)f(a)h(set)g(of)g(the)g(form)g Fh(\031)2385 3650 y Fe(\000)p Fk(1)2381 3711 y Fj(n)2491 3686 y Fl(\(star)o(\()p Fh(v)t Fl(\)\))f(where)i Fh(v)i Fl(is)d(a)f(v)m(ertex)150 3806 y(in)k(the)h(triangulation)e(of)i(\006)1258 3821 y Fj(n)1354 3806 y Fl(\(resp.)73 b(\006)1743 3821 y Fj(n)p Fk(+1)1898 3806 y Fl(\).)e(W)-8 b(e)43 b(de\014ne)h(a)e(v)m(ertex)g (map)g(from)h(\006)3464 3821 y Fj(n)p Fk(+1)3661 3806 y Fl(to)150 3925 y(\006)222 3940 y Fj(n)320 3925 y Fl(b)m(y)h(mapping)g (eac)m(h)h(v)m(ertex)f Fh(u)g Fl(in)f(the)i(triangulation)d(of)i(\006) 2651 3940 y Fj(n)p Fk(+1)2850 3925 y Fl(to)g(a)f(v)m(ertex)h Fh(v)k Fl(in)c(the)150 4045 y(triangulation)22 b(of)h(\006)913 4060 y Fj(n)990 4045 y Fl(where)h Fh(\031)1329 4002 y Fe(\000)p Fk(1)1325 4071 y Fj(n)p Fk(+1)1480 4045 y Fl(\(star)o(\()p Fh(u)p Fl(\)\))j Ff(\032)h Fh(\031)2052 4009 y Fe(\000)p Fk(1)2048 4069 y Fj(n)2159 4045 y Fl(\(star)o(\()p Fh(v)t Fl(\)\).)40 b(After)23 b(extending)g(this)g(v)m(ertex)150 4164 y(map)32 b(linearly)f(to)g(the)i(in)m(terior)e(of)i(higher)f(lev)m (el)g(simplices,)g(the)g(result)h(is)e(our)h(b)s(onding)h(map)150 4284 y Fh(f)199 4299 y Fj(n)281 4284 y Fl(:)27 b(\006)408 4299 y Fj(n)p Fk(+1)591 4284 y Ff(!)g Fl(\006)790 4299 y Fj(n)844 4284 y Fl(.)44 b(Although)33 b(the)g(v)m(ertex)f(map,)h(and) g(therefore)h Fh(f)2737 4299 y Fj(n)2791 4284 y Fl(,)f(ma)m(y)f(not)h (b)s(e)g(uniquely)150 4403 y(de\014ned,)i(the)e(induced)i(c)m(hain)f (map)g Fh(f)1608 4367 y Fe(\003)1597 4428 y Fj(n)1686 4403 y Fl(on)f(cohomology)g(is)g(alw)m(a)m(ys)g(the)g(same)h([BoT].)349 4523 y(T)-8 b(o)30 b(de\014ne)i(p)s(ositivit)m(y)c(recall)i(that)g(w)m (e)h(used)g(an)g(orien)m(tation)e(in)i Fg(R)2887 4487 y Fk(2)2962 4523 y Fl(to)e(de\014ne)j(an)f(orien-)150 4643 y(tation)i(of)i(all)e(2-simplices)i(in)g(\006)1394 4658 y Fj(n)1448 4643 y Fl(.)47 b(Assume)35 b Fh(\013)g Fl(and)f Fh(\014)40 b Fl(are)35 b(t)m(w)m(o)f(suc)m(h)i(orien)m(ted)f (2-simplicies)150 4762 y(whic)m(h)43 b(b)s(oth)f(o)s(ccur)h(within)f (the)h(same)f(tile)g Fh(T)57 b Ff(2)43 b Fl(\006)2201 4777 y Fj(n)2255 4762 y Fl(.)71 b(Then)43 b(the)g(duals)f(to)g Fh(\013)g Fl(and)h Fh(\014)48 b Fl(are)150 4882 y(cohomologous.)65 b(An)m(y)40 b(p)s(ositiv)m(e)g(sum)h(of)f(these)h(cohomology)f(classes) h(w)m(as)f(declared)i(to)e(b)s(e)150 5001 y(P-p)s(ositiv)m(e)33 b(in)g(section)g(3.)349 5121 y(T)-8 b(o)49 b(sho)m(w)h(that)e(this)i (notion)e(of)i(P-p)s(ositivit)m(y)d(is)i(in)m(v)-6 b(arian)m(t)49 b(w)m(e)h(m)m(ust)f(consider)i(the)150 5240 y(isomorphism)37 b(of)864 5215 y(\024)853 5240 y(Cec)m(h)h(cohomology)d(induced)k(b)m(y) d(a)h(homeomorphism)g Fh(h)p Fl(.)55 b(Let)37 b Ff(f)p Fl(\006)3449 5255 y Fj(n)3503 5240 y Ff(g)f Fl(and)1900 5519 y(21)p eop %%Page: 22 23 22 22 bop 150 100 a Ff(f)p Fl(\006)272 63 y Fe(0)272 124 y Fj(n)326 100 y Ff(g)41 b Fl(b)s(e)h(sequences)h(of)e(complexes)h (corresp)s(onding)g(to)f(the)g(tiling)g(spaces)h(and)g(let)f Ff(fA)3647 115 y Fj(n)3700 100 y Ff(g)150 219 y Fl(and)49 b Ff(fA)489 183 y Fe(0)489 244 y Fj(n)543 219 y Ff(g)f Fl(b)s(e)h(sequences)h(of)f(op)s(en)g(co)m(v)m(ers,)k(of)c Fh(X)2200 234 y Fj(x)2298 219 y Fl(and)g Fh(X)2598 183 y Fe(0)2590 244 y Fj(x)2640 219 y Fl(,)j(resp.,)h(corresp)s(onding)c (to)150 339 y(triangulations)36 b(of)i(the)g(complexes.)57 b(Let)37 b Ff(B)1857 354 y Fj(n)1947 339 y Fl(=)e Fh(h)2116 303 y Fe(\000)p Fk(1)2223 339 y Fl(\()p Ff(A)2342 303 y Fe(0)2342 363 y Fj(n)2396 339 y Fl(\).)56 b(F)-8 b(rom)37 b(their)g(construction,)i(for)150 458 y(ev)m(ery)24 b Fh(n)g Fl(there)h(is)f(an)g Fh(m)h Fl(suc)m(h)g(that)f Ff(A)1568 473 y Fj(m)1667 458 y Fl(re\014nes)i Ff(B)2034 473 y Fj(n)2089 458 y Fl(.)41 b(Th)m(us)25 b(w)m(e)f(ma)m(y)g(de\014ne) i(the)e(b)s(onding)h(map)150 578 y Fh(f)199 593 y Fe(B)252 603 y Fc(n)300 593 y Fe(A)364 603 y Fc(m)463 578 y Fl(:)i(\006)590 593 y Fj(m)694 578 y Ff(!)g Fl(\006)893 542 y Fe(0)893 602 y Fj(n)973 578 y Fl(b)m(y)g(mapping)g(eac)m(h)g(v)m(ertex)f Fh(u)g Fl(to)g(a)h(v)m(ertex)f Fh(v)k Fl(where)d(\()p Fh(\031)3045 593 y Fj(m)3120 578 y Fl(\))3159 542 y Fe(\000)p Fk(1)3266 578 y Fl(\(star\()p Fh(u)p Fl(\)\))g Ff(\032)150 698 y Fh(h)207 662 y Fe(\000)p Fk(1)314 698 y Fl(\()p Fh(\031)414 662 y Fe(0)410 723 y Fj(n)464 648 y Fe(\000)p Fk(1)571 698 y Fl(\(star\()p Fh(v)t Fl(\)\)\).)44 b(By)33 b(taking)g(limits)g(w)m(e)h(obtain)f(an)h(isomorphism)h(b)s(et)m(w)m (een)g(the)3548 673 y(\024)3537 698 y(Cec)m(h)150 818 y(cohomology)d(of)i Fh(X)892 833 y Fj(x)975 818 y Fl(and)f(the)1351 792 y(\024)1340 818 y(Cec)m(h)h(cohomology)e(of)i Fh(X)2336 781 y Fe(0)2328 842 y Fj(x)2377 818 y Fl(.)150 997 y Fm(Theorem)28 b(3.)41 b Fd(If)28 b Fh(X)937 1012 y Fj(x)1014 997 y Fd(and)g Fh(X)1291 961 y Fe(0)1283 1022 y Fj(x)1360 997 y Fd(ar)-5 b(e)27 b(two)h(\014xe)-5 b(d-orientation)26 b(tiling)h(sp)-5 b(ac)g(es)27 b(with)g(\014nite)h(r)-5 b(elative)150 1117 y(orientation)37 b(gr)-5 b(oups,)39 b(and)f Fh(h)c Fl(:)e Fh(X)1447 1132 y Fj(x)1530 1117 y Ff(!)g Fh(X)1753 1080 y Fe(0)1745 1141 y Fj(x)1833 1117 y Fd(is)38 b(a)g(home)-5 b(omorphism,)39 b(then)f(the)h (isomorphism)150 1236 y Fh(h)207 1200 y Fe(\003)281 1236 y Fl(:)27 b Fh(H)427 1200 y Fk(2)472 1236 y Fl(\()p Fh(X)602 1200 y Fe(0)594 1261 y Fj(x)643 1236 y Fl(\))h Ff(!)f Fh(H)928 1200 y Fk(2)973 1236 y Fl(\()p Fh(X)1095 1251 y Fj(x)1144 1236 y Fl(\))35 b Fd(pr)-5 b(eserves)35 b(the)g(P-p)-5 b(ositive)33 b(c)-5 b(ones,)35 b(up)h(to)f(an)h(over)-5 b(al)5 b(l)36 b(sign.)150 1416 y Fl(Pro)s(of:)41 b(T)-8 b(o)28 b(see)h(that)f(the)h(P-p)s(ositiv)m(e)f(cone)h(is)f(preserv)m (ed)i(b)m(y)f Fh(h)2547 1380 y Fe(\003)2593 1416 y Fl(,)g(\014rst)g (consider)g(whether)h(the)150 1535 y(map)37 b Fh(h)d Fl(:)f Fh(X)610 1550 y Fj(x)693 1535 y Ff(!)h Fh(X)918 1499 y Fe(0)910 1560 y Fj(x)996 1535 y Fl(is)j(orien)m(tation)f (preserving)h(or)g(rev)m(ersing)g(on)g(eac)m(h)h(path-comp)s(onen)m(t.) 150 1655 y(That)j(is,)h(\014x)f(a)g(tiling)f Fh(x)1109 1670 y Fk(0)1194 1655 y Ff(2)h Fh(X)1384 1670 y Fj(x)1475 1655 y Fl(and)g Fh(h)p Fl(\()p Fh(x)1829 1670 y Fk(0)1874 1655 y Fl(\))f Ff(2)h Fh(X)2151 1619 y Fe(0)2143 1679 y Fj(x)2193 1655 y Fl(.)67 b(There)42 b(is)f(a)g(1:1)f(map)h Fh(j)47 b Fl(:)40 b Fg(R)3408 1619 y Fk(2)3493 1655 y Ff(!)g Fg(R)3705 1619 y Fk(2)150 1774 y Fl(de\014ned)k(b)m(y)e Fh(h)p Fl(\()p Fh(g)t Fl(\()p Fh(x)893 1789 y Fk(0)937 1774 y Fl(\)\))g(=)h Fh(j)6 b Fl(\()p Fh(g)t Fl(\))p Fh(h)p Fl(\()p Fh(x)1507 1789 y Fk(0)1550 1774 y Fl(\))42 b(where)h Fh(g)j Ff(2)d Fg(R)2203 1738 y Fk(2)2290 1774 y Fl(acts)f(b)m(y)h(translation.)70 b(W)-8 b(e)42 b(sa)m(y)g Fh(h)h Fl(is)150 1894 y(orien)m(tation)34 b(preserving)i(or)e(rev)m (ersing)h(based)h(on)f(whether)h Fh(j)k Fl(is)35 b(orien)m(tation)f (preserving)h(or)150 2013 y(rev)m(ersing)k(as)g(a)g(homeomorphism)g(of) g Fg(R)1738 1977 y Fk(2)1783 2013 y Fl(.)60 b(Since)40 b(the)f(orbit)f(of)h(eac)m(h)h(p)s(oin)m(t)e(is)h(dense,)i(and)150 2133 y(since)47 b(the)f(action)g(of)g Fg(R)1101 2097 y Fk(2)1192 2133 y Fl(is)g(con)m(tin)m(uous,)k(whether)d Fh(h)g Fl(is)f(orien)m(tation)g(preserving)g(or)g(not)150 2253 y(is)d(indep)s(enden)m(t)j(of)e(the)g(c)m(hoice)g(of)g Fh(x)1628 2268 y Fk(0)1672 2253 y Fl(.)75 b(Without)44 b(loss)f(of)h(generalit)m(y)-8 b(,)45 b(assume)f(that)f Fh(h)h Fl(is)150 2372 y(orien)m(tation)33 b(preserving.)349 2492 y(The)d(goal)e(is)h(to)f(sho)m(w)i(that)e(ev)m(ery)h(P-p)s(ositiv) m(e)g(cohomology)e(class)j(on)f Fh(X)3143 2456 y Fe(0)3199 2492 y Fl(pulls)g(bac)m(k)g(to)150 2611 y(a)i(P-p)s(ositiv)m(e)h (cohomology)e(class)i(on)g Fh(X)8 b Fl(.)43 b(But)31 b(the)h(P-p)s(ositiv)m(e)f(classes)i(on)f Fh(X)3152 2575 y Fe(0)3211 2611 y Fl(are)g(precisely)150 2731 y(the)26 b(classes)h(represen)m(ted)i(b)m(y)d(p)s(ositiv)m(e)f(linear)h(com)m (binations)g(of)h(basis)f(co)s(c)m(hains)h(on)f(\006)3447 2695 y Fe(0)3447 2755 y Fj(n)3501 2731 y Fl(,)h(with)150 2850 y Fh(n)g Fl(arbitrarily)e(large.)41 b(So)27 b(w)m(e)g(tak)m(e)g(a) f(basic)i(elemen)m(tary)f(co)s(c)m(hain)g(on)g(\006)2852 2814 y Fe(0)2852 2875 y Fj(n)2933 2850 y Fl(\(with)f Fh(n)h Fl(large\),)g(pull)150 2970 y(it)38 b(bac)m(k)h(to)f Fh(X)8 b Fl(,)39 b(and)g(try)f(to)h(sho)m(w)g(that)f(it's)g(p)s(ositiv) m(e)g(there.)61 b(Without)39 b(loss)g(of)g(generalit)m(y)150 3089 y(w)m(e)e(can)f(assume)h(that)f(this)g(co)s(c)m(hain,)h(whic)m(h)g (w)m(e)g(call)e(1)2312 3104 y Fk(\001)2383 3089 y Fl(,)h(is)g(dual)g (to)g(a)g(triangle)g(\001)g(sitting)150 3209 y(in)31 b(the)h(middle)f(of)g(one)h(of)f(the)h(tiles)f(of)g(\006)1723 3173 y Fe(0)1723 3234 y Fj(n)1777 3209 y Fl(.)43 b(W)-8 b(e)32 b(then)g(\014nd)g(an)f Fh(m)h Fl(large)f(enough)h(so)f(that)g (the)150 3329 y(op)s(en)d(co)m(v)m(er)g(induced)i(b)m(y)d(the)h (partition)f(of)h(\006)1891 3344 y Fj(m)1994 3329 y Fl(is)g(a)g (re\014nemen)m(t)h(of)f(the)g(op)s(en)g(co)m(v)m(er)g(induced)150 3448 y(b)m(y)k(the)h(partition)e(of)i(\006)1057 3412 y Fe(0)1057 3473 y Fj(n)1111 3448 y Fl(.)44 b(This)32 b(re\014nemen)m(t)j(giv)m(es)d(a)g(v)m(ertex)g(map)g Fh(\032)g Fl(from)g(the)h(nerv)m(e)g(of)g(the)150 3568 y(co)m(v)m(er)h(of)f(\006)593 3583 y Fj(m)702 3568 y Fl(\(that)f(is,)h(\006)1156 3583 y Fj(m)1265 3568 y Fl(itself)8 b(\))32 b(to)h(the)g(nerv)m(e)h(of)g(the)f(co)m(v)m(er)h(of)f(\006)2839 3531 y Fe(0)2839 3592 y Fj(n)2926 3568 y Fl(\(that)g(is,)f(\006)3380 3531 y Fe(0)3380 3592 y Fj(n)3468 3568 y Fl(itself)8 b(\).)150 3687 y(W)-8 b(e)34 b(m)m(ust)f(sho)m(w)h(that)f Fh(\013)28 b Fl(=)g Fh(\032)1280 3651 y Fe(\003)1325 3687 y Fl(1)1375 3702 y Fk(\001)1479 3687 y Fl(represen)m(ts)35 b(a)e(P-p)s(ositiv)m(e)g(cohomology)f(class.)349 3807 y(No)m(w)49 b Fh(\013)g Fl(itself)h(is)f(not)g(necessarily)h(a)f(p)s (ositiv)m(e)f(linear)h(com)m(bination)g(of)h(elemen)m(tary)150 3926 y(co)s(c)m(hains.)62 b(There)40 b(will)e(b)s(e)h(some)g(triangles) g Fh(T)53 b Fl(on)39 b(\006)2227 3941 y Fj(m)2342 3926 y Fl(where)g Fh(\013)g Fl(ev)-6 b(aluates)40 b(to)e(+1)h(or)g Ff(\000)p Fl(1,)150 4046 y(and)47 b(others)g(where)g(it)e(ev)-6 b(aluates)47 b(to)f(0.)83 b(W)-8 b(e)47 b(call)e(the)i(triangles)f (where)h(it)f(ev)-6 b(aluates)46 b(to)150 4165 y Ff(\006)p Fl(1)39 b Fd(essential)p Fl(.)60 b(Our)40 b(task)e(is)h(to)g(sho)m(w)g (that)g(the)g(p)s(ositiv)m(e)f(essen)m(tial)i(triangles)e(outn)m(um)m (b)s(er)150 4285 y(the)j(negativ)m(e)f(ones)i(in)e(the)h(follo)m(wing)f (sense:)60 b(there)41 b(exists)g(an)g Fh(N)50 b(>)41 b(m)g Fl(suc)m(h)h(that)e(ev)m(ery)150 4404 y(complete)e(tile)f(in)g (\006)943 4419 y Fj(N)1019 4404 y Fl(,)h(mapp)s(ed)g(to)g(\006)1667 4419 y Fj(m)1780 4404 y Fl(and)g(paired)g(with)f Fh(\013)p Fl(,)h(yields)g(a)f(p)s(ositiv)m(e)g(n)m(um)m(b)s(er.)150 4524 y(Put)d(another)h(w)m(a)m(y)-8 b(,)33 b Fh(\013)h Fl(ev)-6 b(aluates)35 b(p)s(ositiv)m(ely)e(on)h(the)h(image)e(\(in)h (\006)2742 4539 y Fj(m)2818 4524 y Fl(\))g(of)g(ev)m(ery)g(tile)g(of)g (lev)m(el)150 4644 y Fh(N)44 b Fl(in)33 b(a)g(particular)g(tiling)f(in) i Fh(X)8 b Fl(.)349 4763 y(Here)29 b(w)m(e)g(are)f(using)h(the)g(iden)m (ti\014cation)g(b)s(et)m(w)m(een)h(a)e(particular)g(tiling)f(in)i Fh(X)35 b Fl(\(view)m(ed)29 b(as)150 4883 y Fg(R)222 4847 y Fk(2)304 4883 y Fl(with)38 b(appropriate)g(mark)m(ers\))g(and)g (a)f(path-comp)s(onen)m(t)j(in)d Fh(X)8 b Fl(.)57 b(In)38 b(this)g(iden)m(ti\014cation,)150 5002 y(a)44 b(p)s(oin)m(t)f(on)h(a)g (tiling)f(corresp)s(onds)i(to)f(a)f(translate)h(of)g(that)f(tiling)g (with)h(that)f(particular)150 5122 y(p)s(oin)m(t)j(situated)g(at)f(the) i(origin.)81 b(Call)45 b(this)h(space)h Fh(P)60 b Fl(\(for)45 b(path-connected\))j(and)f(let)e Fh(P)3722 5086 y Fe(0)150 5241 y Fl(b)s(e)k(the)h(corresp)s(onding)g(path-comp)s(onen)m(t)g(of)f Fh(X)2122 5205 y Fe(0)2149 5241 y Fl(.)92 b(The)49 b(triangulation)f (of)h(\006)3304 5256 y Fj(m)3428 5241 y Fl(induces)1900 5519 y(22)p eop %%Page: 23 24 23 23 bop 150 100 a Fl(a)49 b(triangulation)f(of)i Fh(P)14 b Fl(,)53 b(with)c(eac)m(h)i(v)m(ertex,)h(edge,)i(or)c(triangle)e(of)i Fh(P)63 b Fl(mapping)50 b(b)m(y)g Fh(\031)3675 115 y Fj(m)150 219 y Fl(to)f(a)g(corresp)s(onding)i(v)m(ertex,)i(edge,)h(or) 49 b(triangle)g(of)h(\006)2391 234 y Fj(m)2467 219 y Fl(.)93 b(There)50 b(are)g(t)m(w)m(o)f(metrics)h(on)150 339 y Fh(P)14 b Fl(,)43 b(namely)f(the)g(Euclidean)h(metric)e(and)i (the)f(metric)g(inherited)g(from)g Fh(X)8 b Fl(.)69 b(F)-8 b(or)42 b(short-to-)150 458 y(mo)s(derate)33 b(distances,)h(these)g (metrics)g(b)s(eha)m(v)m(e)g(essen)m(tially)f(iden)m(tically:)150 638 y Fm(Lemma)k(3)p Fl(.)44 b Fd(Ther)-5 b(e)35 b(exist)f(c)-5 b(onstants)34 b Fh(\017)1678 653 y Fk(1)1723 638 y Fd(,)h Fh(\017)1829 653 y Fk(2)1909 638 y Fd(and)f Fh(M)11 b Fd(,)34 b(with)h Fl(0)28 b Fh(<)g(\017)2713 653 y Fk(1)2786 638 y Fh(<)g(M)45 b Fd(and)35 b Fl(0)27 b Fh(<)h(\017)3448 653 y Fk(2)3494 638 y Fd(,)34 b(such)150 757 y(that)g(any)g(two)g(p)-5 b(oints)33 b(in)h(the)f(same)h(p)-5 b(ath)34 b(c)-5 b(omp)g(onent)33 b(with)h(Euclide)-5 b(an)33 b(distanc)-5 b(e)33 b(in)h Fl(\()p Fh(\017)3515 772 y Fk(1)3559 757 y Fh(;)17 b(M)11 b Fl(\))150 877 y Fd(have)35 b(tiling-sp)-5 b(ac)g(e)34 b(distanc)-5 b(e)34 b(gr)-5 b(e)g(ater)35 b(than)h Fh(\017)1878 892 y Fk(2)1923 877 y Fd(.)150 1056 y Fl(Sk)m(etc)m(h)e(of)f(Pro)s(of:) 43 b(F)-8 b(or)33 b(our)g(\014xed)g(set)g(of)h(p)s(olygonal)d(tiles)i (let)f Fh(K)40 b Fl(b)s(e)33 b(a)g(lo)m(w)m(er)g(b)s(ound)h(on)f(the) 150 1176 y(distance)38 b(b)s(et)m(w)m(een)g(the)f(midp)s(oin)m(t)g(of)g (an)m(y)g(edge)g(of)g(an)m(y)g(tile)f(in)h(an)m(y)g(tiling)f(and)h(an)m (y)g(other)150 1296 y(edge)f(in)g(that)f(tiling,)g(and)h(let)f Fh(\013)g Fl(b)s(e)h(a)g(lo)m(w)m(er)f(b)s(ound)i(of)f(the)g(in)m (terior)f(angles)h(of)f(an)m(y)h(of)g(the)150 1415 y(tiles.)41 b(Let)25 b Fh(N)35 b Fl(b)s(e)25 b(suc)m(h)i(that)d(an)m(y)h(op)s(en)g (ball)f(of)h(radius)g Fh(N)36 b Fl(in)24 b(an)m(y)h(tiling)f(con)m (tains)h(at)f(least)h(one)150 1535 y(tile)g(and)h(all)f(the)h(tiles)g (touc)m(hing)g(it.)41 b(The)26 b(result)h(then)f(follo)m(ws)g(if)g Fh(M)38 b(<)28 b Fl(min)p Ff(f)p Fh(K)r(=)p Fl(2)p Fh(;)41 b Fl(sin\()p Fh(\013=)p Fl(2\))p Ff(g)150 1654 y Fl(and)34 b Fh(\017)384 1669 y Fk(2)457 1654 y Fh(<)28 b(\017)602 1669 y Fk(1)647 1654 y Fh(=)-6 b(N)11 b Fl(.)p 869 1664 60 80 v 150 1834 a Fm(Corollary)32 b(2)p Fl(.)43 b Fd(A)30 b(p)-5 b(ath-c)g(onne)g(cte)g(d)29 b(set)i(of)f(tiling-sp)-5 b(ac)g(e)29 b(diameter)h Fh(\017)2773 1849 y Fk(2)2848 1834 y Fd(or)h(less)f(has)h(Euclide)-5 b(an)150 1953 y(diameter)35 b Fh(\017)606 1968 y Fk(1)687 1953 y Fd(or)h(less.)349 2073 y Fl(W)-8 b(e)34 b(return)g(to)f(the)g(pro)s(of)h(of)f(Theorem)h (3.)349 2192 y(Pic)m(k)48 b Fh(\017)627 2207 y Fk(1)p Fj(;)p Fk(2)783 2192 y Fl(for)h Fh(X)55 b Fl(as)48 b(in)g(the)h(lemma.) 88 b(Since)49 b(our)f(homeomorphism)h Fh(h)g Fl(is)f(b)s(et)m(w)m(een) 150 2312 y(compact)40 b(spaces,)i Fh(h)953 2276 y Fe(\000)p Fk(1)1100 2312 y Fl(is)d(uniformly)h(con)m(tin)m(uous,)i(so)d(there)i (is)e(a)g Fh(\016)44 b Fl(for)c(whic)m(h)g(all)f(sets)h(of)150 2432 y(\(tiling-space\))30 b(diamater)f Fh(\016)35 b Fl(or)30 b(less)g(in)g Fh(X)1750 2395 y Fe(0)1808 2432 y Fl(get)f(mapp)s(ed)i(to)f(sets)h(of)f(\(tiling-space\))g(diameter)150 2551 y Fh(\017)190 2566 y Fk(2)263 2551 y Fl(or)e(less)g(in)g Fh(X)8 b Fl(.)42 b(Then)29 b(pic)m(k)f Fh(n)f Fl(large)h(enough)h(that) e(w)m(e)i(can)f(c)m(ho)s(ose)h(our)f(triangle)f(\001)h(to)g(ha)m(v)m(e) 150 2671 y(Euclidean)44 b(diameter)f Fh(\016)k Fl(or)c(less.)74 b(This)44 b(implies)f(that)g(eac)m(h)h(path-connected)h(comp)s(onen)m (t)150 2790 y(\(PCC\))32 b(of)i Fh(h)643 2754 y Fe(\000)p Fk(1)750 2790 y Fl(\(\001\))f(in)g Fh(X)41 b Fl(will)32 b(ha)m(v)m(e)i(Euclidean)f(diameter)h Fh(\017)2518 2805 y Fk(1)2596 2790 y Fl(or)f(less.)349 2910 y(Our)h(pro)s(of)g(of)f(p)s (ositivit)m(y)f(no)m(w)h(pro)s(ceeds)i(in)e(four)h(steps:)349 3029 y(1.)93 b(Sho)m(w)51 b(that)e(preimages)h(in)g Fh(P)64 b Fl(of)50 b(essen)m(tial)g(triangles)f(app)s(ear)h(in)g(disjoin)m(t)g (and)150 3149 y(lo)s(calized)33 b(clusters,)h(with)f(eac)m(h)h(cluster) g(asso)s(ciated)f(to)g(a)g(sp)s(eci\014c)h(PCC)f(of)h Fh(h)3160 3113 y Fe(\000)p Fk(1)3267 3149 y Fl(\(\001\).)349 3268 y(2.)44 b(Sho)m(w)34 b(that)f Fh(\013)g Fl(applied)h(to)f(eac)m(h) h(complete)f(cluster)h(giv)m(es)f(exactly)f(+1.)349 3388 y(3.)42 b(Sho)m(w)28 b(that)g(the)g(n)m(um)m(b)s(er)h(of)f(complete)g (clusters)g(in)g(a)g(tile)f(\(of)g(an)m(y)h(lev)m(el\))f(is)g(b)s (ounded)150 3508 y(from)34 b(b)s(elo)m(w)f(b)m(y)g(a)g(m)m(ultiple)g (of)h(the)f(area)h(of)f(the)h(tile.)349 3627 y(4.)64 b(Sho)m(w)41 b(that)f Fh(\013)p Fl(,)h(applied)g(to)e(triangles)h (whose)h(clusters)g(are)f(only)f(partially)g(in)h(the)150 3747 y(tile,)31 b(is)g(b)s(ounded)h(in)f(magnitude)h(from)f(ab)s(o)m(v) m(e)g(b)m(y)g(a)g(m)m(ultiple)g(of)g(the)h(p)s(erimeter)f(of)h(the)f (tile.)150 3866 y(\(This)i(con)m(tribution)h(need)g(not)f(b)s(e)h(p)s (ositiv)m(e.\))349 3986 y(Since)42 b(v)m(olume)f(scales)h(faster)f (than)g(p)s(erimeter,)i Fh(\013)e Fl(then)h(ev)-6 b(aluates)41 b(p)s(ositiv)m(ely)f(on)h(all)150 4105 y(tiles)33 b(of)g(su\016cien)m (tly)h(high)f(lev)m(el.)349 4225 y(Step)f(1.)44 b(If)31 b Fh(T)46 b Fl(is)31 b(an)h(essen)m(tial)g(triangle,)f(then)h(its)f(3)h (v)m(ertices)f(get)h(sen)m(t)g(b)m(y)g(the)g(b)s(onding)150 4344 y(map)27 b(to)g(the)h(3)f(v)m(ertices)h(of)f(\001.)42 b(But)27 b(this)h(means)g(that)f(the)g(op)s(en)h(star)g(of)f(eac)m(h)h (of)g(the)g(v)m(ertices)150 4464 y(of)41 b Fh(T)54 b Fl(is)41 b(con)m(tained)g(in)f Fh(h)1129 4428 y Fe(\000)p Fk(1)1277 4464 y Fl(of)h(the)g(op)s(en)g(star)f(of)h(the)f(corresp)s (onding)i(v)m(ertex)e(of)h(\001.)65 b(But)150 4583 y(then)31 b(the)f(in)m(tersection)g(of)h(these)f(three)h(op)s(en)f(stars)g (\(namely)g Fh(T)44 b Fl(itself)8 b(\))29 b(is)h(con)m(tained)h(in)f Fh(h)3643 4547 y Fe(\000)p Fk(1)150 4703 y Fl(of)39 b(the)g(in)m (tersection)g(of)g(the)g(op)s(en)h(stars)e(of)h(the)g(v)m(ertices)g(of) g(\001)g(\(namely)f(\001)h(itself)8 b(\).)60 b(Th)m(us)150 4823 y(the)36 b(op)s(en)g(set)g Fh(T)49 b Fl(is)36 b(con)m(tained)g(in) f(the)h(op)s(en)g(set)g Fh(h)2138 4786 y Fe(\000)p Fk(1)2245 4823 y Fl(\(\001\),)g(and)f(ev)m(ery)h(PCC)f(of)h Fh(T)49 b Fl(m)m(ust)36 b(lie)150 4942 y(in)d(a)g(PCC)g(of)h Fh(h)765 4906 y Fe(\000)p Fk(1)872 4942 y Fl(\(\001\).)349 5062 y(W)-8 b(e)38 b(de\014ne)i(a)d(cluster)i(to)e(b)s(e)h(all)f(the)h (PCCs)g(of)g(essen)m(tial)g(triangles)g(within)f(a)h(sp)s(eci\014c)150 5181 y(PCC)29 b(of)g Fh(h)557 5145 y Fe(\000)p Fk(1)665 5181 y Fl(\(\001\).)42 b(The)30 b(PCCs)f(of)h(\001)f(in)g Fh(P)1789 5145 y Fe(0)1846 5181 y Fl(are)g(disjoin)m(t)h(\(since)g (\001)f(w)m(as)h(c)m(hosen)h(to)d(lie)h(in)h(the)1900 5519 y(23)p eop %%Page: 24 25 24 24 bop 150 100 a Fl(in)m(terior)28 b(of)g(a)g(tile)f(in)h(\006)1032 63 y Fe(0)1032 124 y Fj(n)1086 100 y Fl(\))g(and)h(ha)m(v)m(e)f (Euclidean)h(diameter)f Fh(\016)k Fl(or)c(less.)43 b(This)28 b(implies)g(that)g(the)150 219 y(PCCs)37 b(of)f Fh(h)611 183 y Fe(\000)p Fk(1)719 219 y Fl(\(\001\))g(in)g Fh(P)14 b Fl(,)37 b(and)g(therefore)h(the)f(clusters,)h(are)e(disjoin)m(t)h (and)g(ha)m(v)m(e)g(Euclidean)150 339 y(diameter)c Fh(\017)605 354 y Fk(1)684 339 y Fl(or)g(less.)349 458 y(Step)j(2.)51 b(Let)36 b Fh(D)j Fl(b)s(e)c(a)h(PCC)f(of)h Fh(h)1652 422 y Fe(\000)p Fk(1)1759 458 y Fl(\(\001\),)f(and)h(let)f(\000)h(b)s (e)g(the)g(sum)g(of)g(all)e(the)i(triangles)150 578 y(that)25 b(in)m(tersect)h Fh(D)s Fl(.)42 b(\000)25 b(is)g(one)h(of)g(our)f (clusters,)j(plus)e(a)f(n)m(um)m(b)s(er)i(of)e(comp)s(onen)m(ts)i(of)e (inessen)m(tial)150 697 y(triangles.)51 b(W)-8 b(e)36 b(m)m(ust)h(sho)m(w)f(that)g Fh(\013)p Fl(\(\000\))31 b(=)i(+1.)52 b(W)-8 b(e)36 b(do)g(b)m(y)g(sho)m(wing)g(that,)f(as)h(a)g (simplicial)150 817 y(c)m(hain,)e Fh(\032)p Fl(\()p Fh(\031)589 832 y Fj(m)663 817 y Fl(\(\000\)\))27 b(=)i(\001.)349 936 y(Let)e Ff(f)p Fh(v)619 951 y Fj(i)652 936 y Ff(g)f Fl(b)s(e)g(the)h(collection)f(of)g(all)f(v)m(ertices)i(in)f(\000.)42 b(This)26 b(is)h(precisely)f(the)h(set)f(of)h(v)m(ertices)150 1056 y(whose)37 b(stars)f(con)m(tain)h(p)s(oin)m(ts)f(of)h Fh(D)s Fl(.)53 b(If)37 b Fh(v)1780 1071 y Fj(i)1850 1056 y Fl(is)f(one)h(of)f(the)h(v)m(ertices)g(in)f(this)g(collection)g(then) 150 1176 y Fh(\032)p Fl(\()p Fh(\031)298 1191 y Fj(m)373 1176 y Fl(\()p Fh(v)460 1191 y Fj(i)493 1176 y Fl(\)\))c(m)m(ust)i(b)s (e)g(one)f(of)h(the)f(v)m(ertices,)g Fh(d)1893 1191 y Fk(1)1938 1176 y Fh(;)17 b(d)2035 1191 y Fk(2)2079 1176 y Fh(;)g(d)2176 1191 y Fk(3)2219 1176 y Fl(,)33 b(of)g(\001.)44 b(This)34 b(follo)m(ws)f(since)h(\(star)o(\()p Fh(v)3639 1191 y Fj(i)3673 1176 y Fl(\)\))150 1295 y(con)m(tains)40 b(a)e(p)s(oin)m(t)h(in)g Fh(D)s Fl(,)i(and)e(the)g(image)g(of)g(this)g (p)s(oin)m(t)g(lies)g(in)g(exactly)f(three)i(op)s(en)f(sets)150 1415 y(in)44 b(the)g(op)s(en)g(co)m(v)m(er)g(of)f(\006)1173 1379 y Fe(0)1173 1439 y Fj(n)1228 1415 y Fl(,)i(namely)f(the)g(stars)f (of)h(the)g(three)g(v)m(ertices)g(of)g(\001.)75 b(Since)45 b(the)150 1534 y(co)m(v)m(er)39 b(induced)i(b)m(y)d(the)h (triangulation)f(of)h(\006)1905 1549 y Fj(m)2019 1534 y Fl(is)g(a)g(re\014nemen)m(t)h(of)f(the)g(co)m(v)m(er)g(induced)i(b)m (y)150 1654 y(the)d(triangulation)e(of)i(\006)1118 1618 y Fe(0)1118 1678 y Fj(n)1172 1654 y Fl(,)h(the)f(op)s(en)g(set)g (corresp)s(onding)h(to)e Fh(\031)2648 1669 y Fj(m)2723 1654 y Fl(\()p Fh(v)2810 1669 y Fj(i)2844 1654 y Fl(\))g(is)g(con)m (tained)i(in)f(one)150 1773 y(\(or)c(more\))h(or)f(these)h(three)h(op)s (en)f(sets,)g(and)g(in)g(no)f(others.)49 b(Th)m(us)35 b(if)g Fh(\015)40 b Fl(is)34 b(a)h(2-simplex)f(in)h(\000)150 1893 y(then)g Fh(\032)p Fl(\()p Fh(\031)526 1908 y Fj(n)579 1893 y Fl(\()p Fh(\015)6 b Fl(\)\))33 b(is)h(either)h Ff(\006)p Fl(\001)f(or)h(zero.)47 b(Therefore,)36 b Fh(\032)p Fl(\()p Fh(\031)2371 1908 y Fj(m)2445 1893 y Fl(\(\000\)\))29 b(=)h Fh(k)s Fl(\001)k(for)h(some)g(in)m(teger)f Fh(k)s Fl(.)150 2012 y(W)-8 b(e)33 b(will)f(sho)m(w)i(that)e Fh(k)f Fl(=)d(1)33 b(b)m(y)g(sho)m(wing)h(that)e(the)i(b)s(oundary)f (of)g(\000)g(maps)h(to)e(+1)i(times)f(the)150 2132 y(b)s(oundary)h(of)f (\001.)349 2252 y(The)28 b(b)s(oundary)g(of)g(\000)g(is)g(a)f(closed)h (lo)s(op)f(in)h Fh(X)2056 2267 y Fj(x)2106 2252 y Fl(.)42 b(Let)28 b Fh(x)2406 2267 y Fk(1)2450 2252 y Fh(;)17 b(x)2552 2267 y Fk(2)2596 2252 y Fh(;)g(x)2698 2267 y Fk(3)2770 2252 y Fl(b)s(e)28 b(the)g(three)g(v)m(ertices)g(of)150 2371 y Fh(D)g Fl(\(so)c(that)g Ff(f)p Fh(h)p Fl(\()p Fh(x)822 2386 y Fj(i)856 2371 y Fl(\))p Ff(g)g Fl(are)g(the)h(3)g(v)m (ertices)g(of)f(a)h(PCC)f(of)h(\001\).)41 b(W)-8 b(e)25 b(ma)m(y)f(c)m(ho)s(ose)h(3)g(distinguished)150 2491 y(v)m(ertices)32 b Fh(v)557 2506 y Fk(1)602 2491 y Fh(;)17 b(v)695 2506 y Fk(2)739 2491 y Fh(;)g(v)832 2506 y Fk(3)908 2491 y Fl(on)31 b(the)h(b)s(oundary)g(of)g(\000)g(suc)m(h)h(that)e (star\()p Fh(v)2558 2506 y Fj(i)2591 2491 y Fl(\))g(con)m(tains)h Fh(d)3102 2506 y Fj(i)3166 2491 y Fl(for)g Fh(i)c Fl(=)g(1)p Fh(;)17 b Fl(2)p Fh(;)g Fl(3.)150 2610 y(Because)36 b Fh(x)590 2625 y Fj(i)658 2610 y Fl(is)f(con)m(tained)h(in)f(a)g(unique) h(op)s(en)g(set)f(in)g Ff(B)2325 2625 y Fj(n)2415 2610 y Fl(\(namely)f(that)h(corresp)s(onding)h(to)150 2730 y(the)j Fh(d)379 2745 y Fj(i)412 2730 y Fl(\),)g(w)m(e)g(m)m(ust)g(ha)m (v)m(e)g Fh(\032)p Fl(\()p Fh(\031)1305 2745 y Fj(n)1358 2730 y Fl(\()p Fh(v)1445 2745 y Fj(i)1478 2730 y Fl(\)\))d(=)h Fh(d)1758 2745 y Fj(i)1830 2730 y Fl(for)h(all)g Fh(i)p Fl(.)60 b(No)m(w)38 b(let)g Fh(p)2684 2745 y Fk(12)2807 2730 y Fl(denote)h(the)g(path)g(from)150 2852 y Fh(x)207 2867 y Fk(1)287 2852 y Fl(to)34 b Fh(x)467 2867 y Fk(2)547 2852 y Fl(along)g(the)i(b)s(oundary)f(of)h Fh(D)s Fl(.)49 b(There)36 b(is)f(a)g(p)s(ortion)f(of)i(the)f(b)s(oundary)h(of)3404 2827 y Fi(e)3401 2852 y Fl(\000)f(whic)m(h)150 2971 y(forms)42 b(a)g(path)g(from)f Fh(v)1057 2986 y Fk(1)1144 2971 y Fl(to)g Fh(v)1322 2986 y Fk(2)1409 2971 y Fl(with)g(the)h(prop)s(ert)m (y)g(that)f(the)h(star)g(of)g(eac)m(h)g(v)m(ertex)g(along)150 3091 y(this)36 b(path)h(in)m(tersects)g Fh(p)1077 3106 y Fk(12)1162 3091 y Fl(.)52 b(Since)38 b(the)e(only)g(elemen)m(ts)h(of) f Ff(B)2493 3106 y Fj(m)2606 3091 y Fl(that)g(can)g(con)m(tain)h(p)s (oin)m(ts)f(in)150 3210 y Fh(p)200 3225 y Fk(12)315 3210 y Fl(are)31 b(the)g(op)s(en)g(sets)g(corresp)s(onding)h(to)e Fh(d)1887 3225 y Fk(1)1962 3210 y Fl(and)h Fh(d)2205 3225 y Fk(2)2249 3210 y Fl(,)g(the)g(simplicial)f(map)h Fh(\032)16 b Ff(\016)g Fh(\031)3328 3225 y Fj(m)3435 3210 y Fl(applied)150 3330 y(to)43 b(this)h(p)s(ortion)f(of)h(the)g(b)s (oundary)h(of)f(\000)g(giv)m(es)f(a)h(path)g(that)f(b)s(egins)h(at)g Fh(d)3165 3345 y Fk(1)3209 3330 y Fl(,)i(ends)f(at)e Fh(d)3705 3345 y Fk(2)150 3450 y Fl(and)34 b(is)g(completely)f(con)m (tained)i(in)e(the)i(edge)f(b)s(et)m(w)m(een)h Fh(d)2351 3465 y Fk(1)2429 3450 y Fl(and)f Fh(d)2675 3465 y Fk(2)2720 3450 y Fl(.)45 b(The)35 b(net)f(output)g(of)g(the)150 3569 y(c)m(hain)e(map)f Fh(\032)16 b Ff(\016)i Fh(\031)823 3584 y Fj(m)929 3569 y Fl(applied)31 b(to)f(this)h(path)g(is)g (therefore)h(+1)f(times)g(the)g(edge)h(from)f Fh(d)3431 3584 y Fk(1)3506 3569 y Fl(to)g Fh(d)3678 3584 y Fk(2)3722 3569 y Fl(.)150 3689 y(Applying)40 b(the)g(same)g(reasoning)h(around)f (the)h(en)m(tire)f(b)s(oundary)h(of)f(\000)g(and)h(recalling)e(that)150 3808 y(the)32 b(map)g Fh(h)g Fl(is)f(orien)m(tation-preserving,)i(w)m (e)f(see)g(that)f Fh(\032)19 b Ff(\016)g Fh(\031)2456 3823 y Fj(m)2563 3808 y Fl(applied)32 b(to)f(b)s(oundary)h(of)g(\000)g (is)150 3928 y(exactly)g Fh(@)6 b Fl(\001,)32 b(as)i(required.)349 4047 y(Step)j(3.)52 b(Since)36 b(eac)m(h)h(preimage)f(of)g(an)g(essen)m (tial)h(triangle)e(has)i(\014xed)f(Euclidean)g(area,)150 4167 y(and)e(since)g(eac)m(h)g(PCC)f(of)h Fh(h)1228 4131 y Fe(\000)p Fk(1)1335 4167 y Fl(\(\001\))f(has)h(diameter)g Fh(\017)2163 4182 y Fk(1)2241 4167 y Fl(or)f(less,)h(there)g(is)f(an)h (upp)s(er)g(b)s(ound)g(to)150 4286 y(ho)m(w)41 b(man)m(y)f(essen)m (tial)h(triangles)f(can)h(\014t)f(in)g(a)g(cluster.)66 b(The)41 b(n)m(um)m(b)s(er)h(of)f(clusters)g(is)f(then)150 4406 y(b)s(ounded)d(from)f(b)s(elo)m(w)g(b)m(y)g(the)g(n)m(um)m(b)s(er) i(of)e(PCCs)f(of)i(essen)m(tial)f(triangles,)g(divided)g(b)m(y)g(the) 150 4526 y(maxim)m(um)30 b(n)m(um)m(b)s(er)i(of)f(triangles)f(p)s(er)h (cluster.)44 b(But)30 b(eac)m(h)h(essen)m(tial)g(triangle)f(app)s(ears) h(with)150 4645 y(prescrib)s(ed)36 b(p)s(ositiv)m(e)d(densit)m(y)-8 b(,)34 b(so)g(the)g(n)m(um)m(b)s(er)i(of)e(clusters)h(m)m(ust)f(scale)g (with)g(the)g(area.)46 b(Of)150 4765 y(these)40 b(w)m(e)g(m)m(ust)g (subtract)g(o\013)f(the)g(partial)g(clusters)h(near)f(the)h(b)s (oundary)-8 b(,)40 b(but)g(this)f(scales)150 4884 y(lik)m(e)33 b(the)g(p)s(erimeter.)45 b(See)34 b(Step)g(4,)e(b)s(elo)m(w.)349 5004 y(Step)37 b(4.)51 b(Of)37 b(the)f(previously)f(considered)j (clusters,)f(w)m(e)f(m)m(ust)h(giv)m(e)e(sp)s(ecial)h(treatmen)m(t)150 5123 y(to)k(the)h(ones)h(within)e(a)g(distance)i Fh(\017)1531 5138 y Fk(1)1617 5123 y Fl(of)f(the)g(b)s(oundary)g(of)g(the)g(tile.)65 b(These)42 b(ma)m(y)e(lie)h(only)150 5243 y(partially)36 b(within)h(the)g(tile,)h(and)f(the)h(con)m(tribution)g(of)f(the)h (essen)m(tial)g(triangles)f(within)g(the)1900 5519 y(24)p eop %%Page: 25 26 25 25 bop 150 100 a Fl(tile)37 b(ma)m(y)f(not)i(b)s(e)f(p)s(ositiv)m (e.)55 b(Ho)m(w)m(ev)m(er,)39 b(the)e(area)g(of)h(this)f(b)s(oundary)h (region)f(is)g(at)g(most)g Fh(\017)3705 115 y Fk(1)150 219 y Fl(times)43 b(the)g(p)s(erimeter)g(of)h(the)f(tile.)72 b(Since)44 b(the)f(area)g(of)g(eac)m(h)h(triangle)e(is)h(b)s(ounded)i (from)150 339 y(b)s(elo)m(w,)36 b(the)g(n)m(um)m(b)s(er)h(of)e(essen)m (tial)h(triangles)g(in)f(this)h(region)f(is)g(b)s(ounded)i(b)m(y)f(a)f (m)m(ultiple)h(of)150 458 y(the)e(p)s(erimeter.)p 872 468 60 80 v 150 698 a Fm(6.)51 b(Extension)38 b(to)g Fh(X)1046 713 y Fj(\036)1136 698 y Fm(for)h(arbitrary)f(relativ)m(e)e 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1415 y Fl(\()p Fh(X)2683 1430 y Fk(0)2727 1415 y Fl(\),)g(and)f(b)s(et)m(w)m(een)i Fh(H)3503 1379 y Fk(3)3548 1415 y Fl(\(\006)3659 1430 y Fj(\036)3711 1415 y Fl(\))150 1535 y(and)47 b Fh(H)448 1499 y Fk(2)492 1535 y Fl(\(\006)603 1550 y Fk(0)648 1535 y Fl(\),)h(then)f(giv)m(e)f (a)f(consisten)m(t)i(de\014nition)g(of)f(P-p)s(ositivit)m(y)f(for)h Fh(H)3188 1499 y Fk(3)3232 1535 y Fl(\()p Fh(X)3354 1550 y Fj(\036)3407 1535 y Fl(\).)82 b(This)150 1654 y(notion)33 b(of)g(p)s(ositivit)m(y)f(can)h(b)s(e)h(rephrased)h(directly)e(in)g (the)h(language)f(of)2980 1629 y(\024)2969 1654 y(Cec)m(h)h(cohomology) -8 b(.)349 1774 y(When)35 b(w)m(orking)f(with)f Fh(X)1328 1789 y Fj(\036)1381 1774 y Fl(,)g(our)h(complexes)g(\006)2166 1789 y Fj(n)2254 1774 y Fl(are)g(comprised)h(of)f Fh(S)3074 1738 y Fk(1)3119 1774 y Ff(\002)g Fl(tiles.)45 b(T)-8 b(rian-)150 1893 y(gulations)30 b(of)i(\006)758 1908 y Fj(n)843 1893 y Fl(then)f(con)m(tain)h(v)m(ertices,)f(edges,)h (triangles)f(and)g(tetrahedra.)44 b(Corresp)s(ond-)150 2013 y(ingly)-8 b(,)27 b(our)h(op)s(en)g(co)m(v)m(ers,)i(b)m(y)e(stars) g(of)g(v)m(ertices,)g(ha)m(v)m(e)h(single,)f(double,)i(triple)d(and)h (quadruple)150 2132 y(in)m(tersections.)44 b(W)-8 b(e)31 b(orien)m(t)g(the)g(tetrahedra)g(according)g(to)g(some)g(global)f (orien)m(tation)g(on)h(the)150 2252 y(2-dimensional)39 b(Euclidean)g(group.)61 b(The)39 b(dual)g(to)f(an)m(y)g(t)m(w)m(o)h (3-simplices)g(within)f(an)m(y)h Fh(S)3628 2216 y Fk(1)3673 2252 y Ff(\002)150 2372 y Fl(tile)32 b(are)g(cohomologous.)43 b(W)-8 b(e)33 b(declare)g(that)f(the)g(p)s(ositiv)m(e)g(sum)h(of)f(an)m (y)g(of)h(the)g(duals)f(to)g(an)m(y)150 2491 y(of)38 b(these)h(3-simplices)g(is)f(p)s(ositiv)m(e.)57 b(A)38 b(cohomology)f(class)h(on)g Fh(X)2691 2506 y Fj(\036)2781 2491 y Fl(is)g(p)s(ositiv)m(e)g(if)g(it)f(can)i(b)s(e)150 2611 y(represen)m(ted)d(as)d(the)h(pullbac)m(k)f(from)h(some)f(\006) 1929 2626 y Fj(n)2017 2611 y Fl(of)g(suc)m(h)i(a)e(p)s(ositiv)m(e)f (sum.)150 2790 y Fm(Theorem)k(4)p Fl(.)43 b Fd(If)34 b Fh(X)949 2805 y Fj(\036)1036 2790 y Fd(and)g Fh(X)1319 2754 y Fe(0)1311 2818 y Fj(\036)1398 2790 y Fd(ar)-5 b(e)33 b(two)i(al)5 b(l-orientation)33 b(tiling)h(sp)-5 b(ac)g(es,)33 b(and)h Fh(h)28 b Fl(:)f Fh(X)3407 2805 y Fj(\036)3487 2790 y Ff(!)h Fh(X)3706 2754 y Fe(0)3698 2818 y Fj(\036)150 2913 y Fd(is)37 b(a)h(home)-5 b(omorphism,)38 b(then)f(the)h(isomorphism)f Fh(h)2158 2877 y Fe(\003)2236 2913 y Fl(:)31 b Fh(H)2386 2877 y Fk(3)2431 2913 y Fl(\()p Fh(X)2561 2877 y Fe(0)2553 2938 y Fj(x)2602 2913 y Fl(\))h Ff(!)f Fh(H)2895 2877 y Fk(3)2940 2913 y Fl(\()p Fh(X)3062 2928 y Fj(x)3111 2913 y Fl(\))38 b Fd(pr)-5 b(eserves)36 b(the)150 3033 y(P-p)-5 b(ositive)33 b(c)-5 b(ones,)35 b(up)h(to)f(an)h(over)-5 b(al)5 b(l)36 b(sign.)150 3152 y Fl(Sk)m(etc)m(h)30 b(of)f(Pro)s(of:)41 b(The)29 b(pro)s(of)g(of)g (Theorem)g(3)g(carries)g(o)m(v)m(er)g(with)f(only)g(minor)h(mo)s (di\014cation,)150 3272 y(as)k(sk)m(etc)m(hed)i(b)s(elo)m(w.)349 3391 y(W)-8 b(e)31 b(\014rst)g(consider)g(whether)h Fh(h)e Fl(preserv)m(es)i(or)f(rev)m(erses)g(the)g(orien)m(tations)f(of)h(path) g(com-)150 3511 y(p)s(onen)m(ts)j(of)g(the)g(tilings)e(spaces)j({)e (that)g(is,)g(whether)h(it)f(preserv)m(es)i(or)e(rev)m(erses)h(the)g (orien)m(ta-)150 3630 y(tions)f(of)g(the)g(action)f(of)h(the)h (Euclidean)f(group.)44 b(Without)33 b(loss)g(of)g(generalit)m(y)-8 b(,)32 b(assume)h(that)150 3750 y Fh(h)h Fl(is)f(orien)m (tation-preserving.)349 3869 y(If)28 b(\001)e(is)h(a)g(3-simplex)g (arising)f(from)h(the)g(triangulation)f(of)h(\006)2643 3833 y Fe(0)2643 3894 y Fj(n)2697 3869 y Fl(,)h(with)f Fh(n)f Fl(large,)i(w)m(e)f(c)m(ho)s(ose)150 3989 y Fh(m)41 b Fl(suc)m(h)h(that)f(the)g(op)s(en)g(co)m(v)m(er)h(induced)g(b)m(y)f (the)g(triangulation)f(of)h(\006)2929 4004 y Fj(m)3045 3989 y Fl(re\014nes)h(the)f(op)s(en)150 4109 y(co)m(v)m(er)28 b(induced)g(b)m(y)f(the)h(triangulation)e(of)h(\006)1835 4072 y Fe(0)1835 4133 y Fj(n)1889 4109 y Fl(,)h(pulled)g(bac)m(k)f(to)g Fh(X)2663 4124 y Fj(\036)2743 4109 y Fl(b)m(y)g Fh(h)2932 4072 y Fe(\000)p Fk(1)3039 4109 y Fl(.)42 b(W)-8 b(e)28 b(then)f(de\014ne)150 4228 y(the)35 b(co)s(c)m(hain)g Fh(\013)29 b Fl(=)h Fh(\032)936 4192 y Fe(\003)982 4228 y Fl(1)1032 4243 y Fk(\001)1136 4228 y Fl(on)35 b(\006)1348 4243 y Fj(n)1402 4228 y Fl(,)f(where)h Fh(\032)30 b Fl(:)f(\006)1964 4243 y Fj(m)2069 4228 y Ff(!)g Fl(\006)2270 4192 y Fe(0)2270 4253 y Fj(n)2359 4228 y Fl(is)34 b(a)g(v)m(ertex)g(corresp)s(onding)i (to)e(the)150 4348 y(re\014nemen)m(t.)58 b(W)-8 b(e)38 b(call)e(a)h(tetrahedron)h(on)g(\006)f Fd(essential)f Fl(if)i Fh(\013)f Fl(applied)h(to)e(that)h(tetrahedron)150 4467 y(is)31 b(nonzero.)45 b(W)-8 b(e)31 b(m)m(ust)h(sho)m(w)g(that)f (ev)m(ery)g(tile)g(of)h(su\016cien)m(tly)f(high)h(lev)m(el,)f(crossed)i (with)e Fh(S)3678 4431 y Fk(1)3722 4467 y Fl(,)150 4587 y(con)m(tains)36 b(more)g(preimages)g(of)g(p)s(ositiv)m(e)f(essen)m (tial)i(tetrahedra)f(than)g(of)g(negativ)m(e)f(essen)m(tial)150 4706 y(tetrahedra.)349 4826 y(As)40 b(b)s(efore)h(w)m(e)g(w)m(ork)e(on) h(a)g(particular)g(path-comp)s(onen)m(t)h Fh(P)54 b Fl(of)40 b Fh(X)2984 4841 y Fj(\036)3037 4826 y Fl(.)64 b Fh(P)54 b Fl(is)40 b(naturally)150 4945 y(iden)m(ti\014ed,)48 b(not)d(with)f(a)g(single)h(tiling,)h(but)f(with)f(the)h(unit)g(tangen) m(t)g(bundle)h(of)f(a)f(single)150 5065 y(tiling)38 b(as)h(follo)m(ws.) 62 b(A)39 b(unit)h(v)m(ector)f(at)g(a)g(particular)g(p)s(oin)m(t)g(in)g (the)h(tiling)e(means)i(that)f(the)150 5185 y(tiling)33 b(should)i(b)s(e)f(translated)h(to)e(put)i(that)f(p)s(oin)m(t)g(at)f (the)i(origin,)e(and)i(rotated)f(so)g(that)g(the)1900 5519 y(25)p eop %%Page: 26 27 26 26 bop 150 100 a Fl(v)m(ector)40 b(p)s(oin)m(ts)g(in)h(the)f(p)s (ositiv)m(e)g Fh(x)p Fl(-direction.)65 b(This)40 b(unit)h(tangen)m(t)f (bundle)i(has)e(a)g(natural)150 219 y(Riemannian)30 b(metric)f(that)g (pla)m(ys)h(the)f(same)h(role)f(as)h(that)f(pla)m(y)m(ed)h(b)m(y)f(the) h(Euclidean)g(metric)150 339 y(in)k(the)h(pro)s(of)f(of)h(Theorem)g(3.) 47 b(In)34 b(particular,)g(analogs)g(to)g(Lemma)g(3)g(and)h(Corollary)e (2)h(are)150 458 y(easily)f(pro)m(v)m(en.)349 578 y(The)d(pro)s(of)g (then)g(pro)s(ceeds)h(in)e(4)g(steps,)i(exactly)d(as)i(b)s(efore.)44 b(W)-8 b(e)29 b(sho)m(w)h(that)f(preimages)150 697 y(of)40 b(essen)m(tial)h(tetrahedra)g(app)s(ear)g(in)f(clusters,)i(that)e Fh(\013)h Fl(applied)f(to)g Fh(\031)2895 712 y Fj(m)3011 697 y Fl(of)g(eac)m(h)i(complete)150 817 y(cluster)c(is)f(+1,)h(that)f (the)h(n)m(um)m(b)s(er)h(of)e(complete)h(clusters)g(in)f(a)g(tile)g(of) h(high)f(lev)m(el)g(\(crossed)150 936 y(with)30 b Fh(S)441 900 y Fk(1)486 936 y Fl(\))g(scales)h(as)g(the)f(area)h(of)g(the)f (tile,)g(and)h(that)g(the)f(e\013ect)i(of)e(the)h(incomplete)g (clusters)150 1056 y(is)i(b)s(ounded)i(b)m(y)e(a)g(m)m(ultiple)h(of)f (the)h(p)s(erimeter)f(of)h(the)f(tile.)349 1176 y(The)40 b(only)e(subtlet)m(y)h(is)g(in)g(step)h(2,)g(where)g(previously)e(w)m (e)i(had)f(made)h(explicit)e(use)i(of)150 1295 y(the)32 b(geometry)e(of)i(triangles.)43 b(W)-8 b(e)31 b(de\014ne)i Fh(D)h Fl(and)e(\000)f(as)h(b)s(efore,)g(and)g(sho)m(w)f(that)g(eac)m (h)h(v)m(ertex)150 1415 y(in)j(\000)g(m)m(ust)h(b)s(e)f(mapp)s(ed)h(b)m (y)f Fh(\032)23 b Ff(\016)g Fh(\031)1477 1430 y Fj(m)1587 1415 y Fl(to)34 b(a)h(v)m(ertex)g(of)g(\001.)49 b(Those)36 b(v)m(ertices)f(in)g(\000)g(whose)g(op)s(en)150 1534 y(stars)28 b(con)m(tain)g(a)g(v)m(ertex)f Fh(x)1161 1549 y Fj(i)1222 1534 y Fl(of)h Fh(D)s Fl(,)h(m)m(ust)f(get)g(mapp)s(ed)h (to)e(the)h(v)m(ertex)g Fh(d)2879 1549 y Fj(i)2940 1534 y Fl(of)g(\001.)42 b(Those)28 b(whose)150 1654 y(op)s(en)h(stars)g(in)m (tersect)g(the)g(edge)g(of)g Fh(D)j Fl(that)c(runs)i(from)e Fh(x)2346 1669 y Fj(i)2408 1654 y Fl(to)g Fh(x)2582 1669 y Fj(j)2653 1654 y Fl(m)m(ust)h(get)f(mapp)s(ed)i(to)e Fh(d)3600 1669 y Fj(i)3661 1654 y Fl(or)150 1773 y Fh(d)202 1788 y Fj(j)244 1773 y Fl(.)44 b(Those)34 b(that)f(in)m(tersect)h(the)f (face)h(whose)g(corners)g(are)g Fh(x)2463 1788 y Fj(i)2496 1773 y Fl(,)f Fh(x)2614 1788 y Fj(j)2689 1773 y Fl(and)g Fh(x)2939 1788 y Fj(k)3022 1773 y Fl(m)m(ust)g(b)s(e)h(mapp)s(ed)150 1893 y(to)f Fh(d)324 1908 y Fj(i)357 1893 y Fl(,)g Fh(d)470 1908 y Fj(j)546 1893 y Fl(or)g Fh(d)720 1908 y Fj(k)769 1893 y Fl(.)45 b(These)35 b(considerations,)f(together)f(with)h(the)g (fact)g(that)f(\000)h(is)f(con)m(tractible,)150 2012 y(imply)h(that)g Fh(@)6 b Fl(\000,)34 b(view)m(ed)h(as)f(a)g(c)m(hain,) i(is)e(mapp)s(ed)h(to)f(+1)h(times)f Fh(@)6 b Fl(\001,)34 b(and)h(hence)h(that)e(\000)h(is)150 2132 y(mapp)s(ed)f(to)f(exactly)f (+1)i(times)f(\001.)p 1636 2142 60 80 v 150 2372 a Fm(7.)51 b(Algebraic)37 b(consequences.)349 2491 y Fl(Belo)m(w)31 b(w)m(e)g(examine)g(some)g(of)g(the)h(consequences)h(of)e(isomorphism)g (of)g(ordered)h(groups)150 2611 y(obtained)25 b(as)f(direct)g(limits.) 40 b(By)24 b(Theorems)h(3)f(and)h(4)f(this)g(will)f(sho)m(w)i(that)f (certain)g(prop)s(erties)150 2730 y(of)46 b(the)h(area)f(stretc)m(hing) g(factor)h(are)f(homeomorphism)h(in)m(v)-6 b(arian)m(ts)46 b(of)g Fh(X)3083 2745 y Fj(x)3179 2730 y Fl(for)g(the)h(\014nite)150 2850 y(orien)m(tation)33 b(case,)g(and)h(are)f(homeomorphism)h(in)m(v) -6 b(arian)m(ts)34 b(of)f Fh(X)2657 2865 y Fj(\036)2743 2850 y Fl(in)g(all)f(cases.)349 2969 y(Our)50 b(setting)e(is)h(as)g (follo)m(ws.)90 b(W)-8 b(e)49 b(assume)g(w)m(e)g(ha)m(v)m(e)g(Ab)s (elian)g(groups)g Fh(H)56 b Fl(and)49 b Fh(H)3694 2933 y Fe(0)3722 2969 y Fl(,)150 3089 y(obtained)28 b(as)f(direct)h(limits)e (of)i(\014nitely)f(generated)h(free)g(Ab)s(elian)f(groups)h(b)m(y)g (primitiv)m(e)f(maps)150 3208 y Fh(\036)209 3172 y Fe(\003)299 3208 y Fl(with)44 b(P)m(erron)g(eigen)m(v)-6 b(alues)45 b Fh(\025)f Fl(and)g Fh(\025)1763 3172 y Fe(0)1791 3208 y Fl(.)76 b(An)44 b(elemen)m(t)h(of)f Fh(H)51 b Fl(will)43 b(b)s(e)h(denoted)h([\()p Fh(v)t(;)17 b(k)s Fl(\)],)150 3328 y(meaning)38 b(the)g(class)f(of)h(the)g(elemen)m(t)g Fh(v)k Fl(in)37 b(the)h Fh(k)s Fl(-th)g(appro)m(ximan)m(t)f(to)g Fh(H)8 b Fl(.)57 b(Of)38 b(course,)i(the)150 3448 y(application)i(w)m (e)g(ha)m(v)m(e)g(in)g(mind)h(is)f(for)g Fh(H)50 b Fl(to)41 b(b)s(e)i(the)f(top)2511 3422 y(\024)2500 3448 y(Cec)m(h)h(cohomology)e (of)h(a)g(tiling)150 3567 y(space,)35 b(for)g(the)g(free)g(Ab)s(elian)f (groups)h(to)f(b)s(e)h(the)g(free)g(part)f(of)h(the)g(top)f(cohomology) f(of)i(\006)3700 3582 y Fj(x)150 3687 y Fl(or)e(\006)344 3702 y Fj(\036)397 3687 y Fl(,)g(and)g(for)h(the)g(primitiv)m(e)e(map)h (to)g(b)s(e)h(induced)h(b)m(y)e(substitution.)150 3866 y Fm(Lemma)46 b(4)p Fl(.)66 b Fd(We)41 b(c)-5 b(an)42 b(cho)-5 b(ose)41 b(the)h(left)g(Perr)-5 b(on)41 b(eigenve)-5 b(ctor)41 b Fh(r)h Fl(=)e(\()p Fh(r)2917 3881 y Fk(1)2961 3866 y Fh(;)17 b(r)3051 3881 y Fk(2)3095 3866 y Fh(;)g(:)g(:)g(:)d(;)j (r)3362 3881 y Fj(n)3416 3866 y Fl(\))42 b Fd(of)f(the)150 3986 y(primitive)34 b(matrix)i Fh(\036)955 3950 y Fe(\003)1037 3986 y Fd(so)f(that)h(e)-5 b(ach)35 b Fh(r)1637 4001 y Fj(i)1706 3986 y Fd(is)g(a)g(p)-5 b(olynomial)37 b(in)e Fh(\025)h Fd(with)f(inte)-5 b(ger)35 b(c)-5 b(o)g(e\016cients.)150 4165 y Fl(Pro)s(of:)44 b(T)-8 b(o)32 b(\014nd)j Fh(r)s Fl(,)d(w)m(e)i(wish)f(to)g(solv)m(e)g(the)h(equation)1614 4424 y Fh(r)s Fl(\()p Fh(I)8 b(\025)21 b Ff(\000)i Fh(\036)1991 4383 y Fe(\003)2037 4424 y Fl(\))k(=)i(0)p Fh(:)1325 b Fl(11\))150 4682 y(In)49 b(order)g(to)f(solv)m(e)g(this,)k(w)m(e)d(p) s(erform)g(column)g(op)s(erations)f(un)m(til)h(the)g(matrix)e Fh(\036)3457 4646 y Fe(\003)3552 4682 y Fl(is)h(in)150 4802 y(lo)m(w)m(er)34 b(triangular)f(form.)46 b(Throughout)35 b(this)f(pro)s(cess,)g(eac)m(h)h(en)m(try)f(of)g(the)g(matrix)f(will)g (b)s(e)h(a)150 4921 y(quotien)m(t)h(of)g(in)m(teger)h(p)s(olynomials)e (in)h Fh(\025)p Fl(.)50 b(Therefore)37 b(there)f(is)f(a)g(solution)g (to)g(the)g(equation)150 5041 y(where)i(eac)m(h)g(en)m(try)f(of)g Fh(r)i Fl(is)e(a)g(quotien)m(t)g(of)g(in)m(teger)g(p)s(olynomials)f(in) h Fh(\025)p Fl(.)52 b(Th)m(us)37 b(b)m(y)f(rescaling)150 5160 y(w)m(e)h(ma)m(y)f(obtain)g(a)g(ro)m(w)h(eigen)m(v)m(ector)g(suc)m (h)g(that)f(eac)m(h)i(en)m(try)e(is)h(an)f(in)m(teger)h(p)s(olynomial)e (in)150 5280 y Fh(\025)p Fl(.)p 340 5290 V 1900 5519 a(26)p eop %%Page: 27 28 27 27 bop 150 100 a Fd(Inte)-5 b(ger)31 b(Case:)41 b Fl(In)29 b(the)g(case)g(where)g Fh(\025)g Fl(is)f(an)h(in)m(teger,)h(w) m(e)f(will)e(tak)m(e)h(the)h Fh(r)i Fl(from)e(the)g(previous)150 219 y(lemma)36 b(and)h(scale)f(b)m(y)g(a)g(factor)g(of)h(1)p Fh(=g)i Fl(where)e Fh(g)i Fl(is)d(the)g(greatest)g(common)h(divisor)e (of)i(the)150 339 y(en)m(tries)d(of)g Fh(r)s Fl(.)150 458 y Fd(Noninte)-5 b(ger)38 b(Case:)51 b Fl(In)38 b(the)f(case)g (where)h Fh(\025)f Fl(is)g(not)g(an)g(in)m(teger)h(\(and)f(therefore)h (irrational\),)150 578 y(tak)m(e)h(the)h Fh(r)i Fl(from)e(the)g (previous)g(lemma)f(and)h(scale)g(b)m(y)g(a)f(factor)h(of)g(1)p Fh(=\025)3036 542 y Fj(p)3121 578 y Fl(where)g Fh(p)g Fl(is)g(the)150 697 y(maximal)34 b(p)s(o)m(w)m(er)i(of)g Fh(\025)g Fl(that)f(o)s(ccurs)i(in)e(the)h(en)m(tries)h(of)f Fh(r)s Fl(.)50 b(No)m(w)36 b(the)g(ro)m(w)f(eigen)m(v)m(ector)h(is)g (in)150 817 y Fg(Z)p Fl([1)p Fh(=\025)p Fl(].)349 936 y(W)-8 b(e)34 b(de\014ne)h(the)f(map)g Fh(\026)28 b Fl(:)g Fh(H)36 b Ff(!)27 b Fg(R)34 b Fl(b)m(y)f Fh(\026)p Fl([\()p Fh(v)t(;)17 b(k)s Fl(\)])26 b Ff(7!)i Fh(\025)2395 900 y Fe(\000)p Fj(k)2507 936 y Fh(r)s(v)37 b Fl(where)d Fh(r)i Fl(is)e(\014xed)g(as)f(ab)s(o)m(v)m(e.)150 1056 y(In)f(either)h(the)f(in)m(teger)h(or)f(nonin)m(teger)h(case)f(it)g(is) g(clear)g(that)g(the)g(image)g(of)g Fh(\026)g Fl(is)g(a)g(subset)i(of) 150 1176 y Fg(Z)p Fl([1)p Fh(=\025)p Fl(].)43 b(Our)35 b(p)s(ositiv)m(e)d(cone)i Fh(H)1385 1191 y Fk(+)1484 1176 y Fl(is)f(then)h(the)g(preimage)f(of)h(the)f(p)s(ositiv)m(e)g (real)g(n)m(um)m(b)s(ers.)349 1295 y(No)m(w)k(w)m(e)g(supp)s(ose)i (that)e(w)m(e)g(ha)m(v)m(e)h(t)m(w)m(o)f(suc)m(h)h(groups,)g Fh(H)45 b Fl(and)38 b Fh(H)2904 1259 y Fe(0)2932 1295 y Fl(,)f(with)g(maps)g Fh(\026)g Fl(and)150 1415 y Fh(\026)210 1379 y Fe(0)280 1415 y Fl(from)43 b(eac)m(h)g(to)f Fg(R)p Fl(,)i(constructed)g(as)e(ab)s(o)m(v)m(e,)j(with)d(P)m(erron)h(eigen)m (v)-6 b(alues)43 b Fh(\025)f Fl(and)h Fh(\025)3489 1379 y Fe(0)3517 1415 y Fl(,)h(and)150 1534 y(supp)s(ose)32 b(that)f Fh(h)d Fl(:)f(\()p Fh(H)s(;)17 b(H)1129 1549 y Fk(+)1194 1534 y Fl(\))27 b Ff(!)g Fl(\()p Fh(H)1517 1498 y Fe(0)1545 1534 y Fh(;)17 b(H)1681 1498 y Fe(0)1673 1559 y Fk(+)1738 1534 y Fl(\))31 b(is)f(an)h(order)h(isomorphism.)43 b(\(W)-8 b(e)31 b(do)g(not)g(assume)150 1654 y(that)f Fh(h)g Fl(resp)s(ects)h(the)g(direct)f(limit)f(structure,)i(only)e (that)h(it)f(resp)s(ects)j(the)e(order)h(structure.\))150 1833 y Fm(Lemma)38 b(5)p Fl(.)44 b Fd(L)-5 b(et)36 b Fh(x;)17 b(y)31 b Ff(2)d Fh(H)43 b Fd(with)36 b Fh(\026)p Fl(\()p Fh(x)p Fl(\))f Fd(and)g Fh(\026)p Fl(\()p Fh(y)t Fl(\))g Fd(b)-5 b(oth)35 b(p)-5 b(ositive.)44 b(Then)1621 2049 y Fh(\026)p Fl(\()p Fh(x)p Fl(\))p 1621 2094 195 4 v 1623 2185 a Fh(\026)p Fl(\()p Fh(y)t Fl(\))1855 2117 y(=)1972 2049 y Fh(\026)2032 2013 y Fe(0)2060 2049 y Fh(h)p Fl(\()p Fh(x)p Fl(\))p 1972 2094 280 4 v 1974 2185 a Fh(\026)2034 2156 y Fe(0)2062 2185 y Fh(h)p Fl(\()p Fh(y)t Fl(\))2264 2117 y Fh(:)1320 b Fl(12\))150 2455 y(Pro)s(of:)44 b(Let)33 b Fh(a;)17 b(b)32 b Fl(b)s(e)i(p)s(ositiv)m(e)e (in)m(tegers.)45 b(Then)1113 2676 y Fh(ax)21 b Ff(\000)i Fh(by)31 b Ff(2)d Fh(H)1644 2691 y Fk(+)1765 2676 y Ff(\()-17 b(\))55 b Fh(\026)p Fl(\()p Fh(ax)22 b Ff(\000)g Fh(by)t Fl(\))27 b Ff(\025)h Fl(0)1765 2825 y Ff(\()-17 b(\))55 b Fh(a\026)p Fl(\()p Fh(x)p Fl(\))21 b Ff(\000)i Fh(b\026)p Fl(\()p Fh(y)t Fl(\))j Ff(\025)j Fl(0)1765 3032 y Ff(\()-17 b(\))2015 2964 y Fh(\026)p Fl(\()p Fh(x)p Fl(\))p 2015 3009 195 4 v 2017 3100 a Fh(\026)p Fl(\()p Fh(y)t Fl(\))2249 3032 y Ff(\025)2371 2964 y Fh(b)p 2366 3009 53 4 v 2366 3100 a(a)2431 3032 y(:)3612 2883 y Fl(13\))150 3310 y(Since)34 b Fh(h)g Fl(is)f(an)g(order)h(isomorphism,)1071 3531 y Fh(ax)22 b Ff(\000)h Fh(by)30 b Ff(2)e Fh(H)1602 3546 y Fk(+)1724 3531 y Ff(\()-17 b(\))55 b Fh(ah)p Fl(\()p Fh(x)p Fl(\))21 b Ff(\000)i Fh(bh)p Fl(\()p Fh(y)t Fl(\))k Ff(2)h Fh(H)2771 3490 y Fe(0)2763 3555 y Fk(+)1724 3738 y Ff(\()-17 b(\))1974 3670 y Fh(\026)2034 3634 y Fe(0)2062 3670 y Fh(h)p Fl(\()p Fh(x)p Fl(\))p 1974 3715 280 4 v 1976 3806 a Fh(\026)2036 3777 y Fe(0)2064 3806 y Fh(h)p Fl(\()p Fh(y)t Fl(\))2293 3738 y Ff(\025)2415 3670 y Fh(b)p 2410 3715 53 4 v 2410 3806 a(a)2475 3738 y(:)3612 3663 y Fl(14\))150 4022 y(Therefore)32 b Fh(b=a)26 b Ff(\024)j Fh(\026)p Fl(\()p Fh(x)p Fl(\))p Fh(=\026)p Fl(\()p Fh(y)t Fl(\))f(if)i(and)h(only)f(if)g Fh(b=a)d Ff(\024)h Fh(\026)2262 3986 y Fe(0)2290 4022 y Fh(h)p Fl(\()p Fh(x)p Fl(\))p Fh(=\026)2592 3986 y Fe(0)2619 4022 y Fh(h)p Fl(\()p Fh(y)t Fl(\).)43 b(Since)31 b(this)f(is)g(true)h (for)150 4142 y(an)m(y)i(p)s(ositiv)m(e)g(in)m(tegers)h Fh(a)f Fl(and)g Fh(b)g Fl(it)g(m)m(ust)h(b)s(e)f(the)h(case)g(that)1556 4352 y Fh(\026)1616 4316 y Fe(0)1644 4352 y Fh(h)p Fl(\()p Fh(x)p Fl(\))p 1556 4396 280 4 v 1558 4488 a Fh(\026)1618 4459 y Fe(0)1646 4488 y Fh(h)p Fl(\()p Fh(y)t Fl(\))1875 4419 y(=)1992 4352 y Fh(\026)p Fl(\()p Fh(x)p Fl(\))p 1992 4396 195 4 v 1994 4488 a Fh(\026)p Fl(\()p Fh(y)t Fl(\))2199 4419 y Fh(:)p 2286 4429 60 80 v 1385 w Fl(15\))150 4764 y Fm(Lemma)k(6)p Fl(.)44 b Fg(Q)p Fl([)p Fh(\025)p Fl(])36 b Fd(and)g Fg(Q)p Fl([)p Fh(\025)1269 4728 y Fe(0)1297 4764 y Fl(])f Fd(ar)-5 b(e)35 b(identic)-5 b(al)35 b(as)g(subsets)h(of)f Fg(R)p Fd(.)150 4943 y Fl(Pro)s(of:)44 b(First)33 b(w)m(e)g(claim)g(that)g(there)h(exist)e Fh(p)p Fl(\()p Fh(x)p Fl(\))p Fh(;)17 b(q)t Fl(\()p Fh(x)p Fl(\))26 b Ff(2)j Fg(Z)p Fl([)p Fh(x)p Fl(])j(suc)m(h)j(that)1722 5227 y Fh(\025)28 b Fl(=)1925 5159 y Fh(p)p Fl(\()p Fh(\025)2072 5123 y Fe(0)2100 5159 y Fl(\))p 1925 5204 214 4 v 1926 5295 a Fh(q)t Fl(\()p Fh(\025)2071 5266 y Fe(0)2099 5295 y Fl(\))2150 5227 y Fh(:)1434 b Fl(16\))1900 5519 y(27)p eop %%Page: 28 29 28 28 bop 150 100 a Fl(Cho)s(ose)33 b Fh(x)28 b Fl(=)g([\()p Fh(v)t(;)17 b(k)s Fl(\)])26 b Ff(2)i Fh(H)41 b Fl(suc)m(h)35 b(that)e Fh(\026)p Fl(\()p Fh(x)p Fl(\))27 b Fh(>)h Fl(0.)44 b(Let)33 b Fh(y)e Fl(=)d([\()p Fh(v)t(;)17 b(k)24 b Fl(+)f(1\)])k Ff(2)h Fh(H)8 b Fl(.)44 b(Then)1525 322 y Fh(\026)1585 286 y Fe(0)1613 322 y Fh(h)p Fl(\()p Fh(x)p Fl(\))p 1525 366 280 4 v 1527 458 a Fh(\026)1587 429 y Fe(0)1615 458 y Fh(h)p Fl(\()p Fh(y)t Fl(\))1845 389 y(=)1962 322 y Fh(\026)p Fl(\()p Fh(x)p Fl(\))p 1962 366 195 4 v 1964 458 a Fh(\026)p Fl(\()p Fh(y)t Fl(\))2196 389 y(=)28 b Fh(\025:)1225 b Fl(17\))150 680 y(Since)34 b(the)g(image)f(of)g Fh(\026)1043 644 y Fe(0)1104 680 y Fl(is)g(a)g(subset)i(of)e Fg(Z)p Fl([1)p Fh(=\025)1958 644 y Fe(0)1986 680 y Fl(],)g(the)g(claim) g(follo)m(ws.)349 799 y(No)m(w)28 b(let)f Fh(f)11 b Fl(\()p Fh(x)p Fl(\))27 b(b)s(e)h(the)h(minimal)e(monic)h(p)s(olynomial)e(in)i Fg(Z)p Fl([)p Fh(x)p Fl(])g(for)g(the)g(algebraic)g(in)m(teger)150 919 y Fh(\025)208 883 y Fe(0)236 919 y Fl(.)91 b(Since)50 b Fh(q)t Fl(\()p Fh(\025)776 883 y Fe(0)804 919 y Fl(\))j Ff(6)p Fl(=)h(0,)f(the)c(p)s(olynomial)e Fh(f)11 b Fl(\()p Fh(x)p Fl(\))48 b(do)s(es)h(not)g(divide)g Fh(q)t Fl(\()p Fh(x)p Fl(\).)90 b(Since)50 b Fh(f)11 b Fl(\()p Fh(x)p Fl(\))48 b(is)150 1039 y(irreducible,)32 b(the)g(n)m(um)m(b)s(er)g(1)f (is)g(a)g(greatest)g(common)h(divisor)e(of)h Fh(f)11 b Fl(\()p Fh(x)p Fl(\))30 b(and)i Fh(q)t Fl(\()p Fh(x)p Fl(\))f(in)g(the)g(ring)150 1158 y Fg(Q)p Fl([)p Fh(x)p Fl(].)349 1278 y(Since)d Fg(Q)p Fl([)p Fh(x)p Fl(])e(is)g(a)h (principal)f(ideal)h(domain,)g(there)g(exist)g(p)s(olynomials)e Fh(a)p Fl(\()p Fh(x)p Fl(\))p Fh(;)17 b(b)p Fl(\()p Fh(x)p Fl(\))25 b Ff(2)j Fg(Q)p Fl([)p Fh(x)p Fl(])150 1397 y(suc)m(h)35 b(that)1414 1520 y Fh(a)p Fl(\()p Fh(x)p Fl(\))p Fh(q)t Fl(\()p Fh(x)p Fl(\))20 b(+)j Fh(b)p Fl(\()p Fh(x)p Fl(\))p Fh(f)11 b Fl(\()p Fh(x)p Fl(\))26 b(=)i(1)p Fh(:)1125 b Fl(18\))150 1712 y(This)33 b(implies)1631 1834 y Fh(a)p Fl(\()p Fh(\025)1781 1793 y Fe(0)1808 1834 y Fl(\))p Fh(q)t Fl(\()p Fh(\025)1992 1793 y Fe(0)2020 1834 y Fl(\))27 b(=)i(1)p Fh(:)1342 b Fl(19\))150 2027 y(Therefore,)1456 2149 y Fh(\025)28 b Fl(=)g Fh(a)p Fl(\()p Fh(\025)1797 2108 y Fe(0)1825 2149 y Fl(\))p Fh(q)t Fl(\()p Fh(\025)2009 2108 y Fe(0)2036 2149 y Fl(\))g Ff(2)g Fg(Q)p Fl([)p Fh(\025)2360 2108 y Fe(0)2388 2149 y Fl(])p Fh(:)1168 b Fl(20\))150 2341 y(whic)m(h)34 b(implies)f(that)g(as)g(sets)h(of)g (real)f(n)m(um)m(b)s(ers,)1665 2583 y Fg(Q)p Fl([)p Fh(\025)p Fl(])27 b Ff(\022)i Fg(Q)p Fl([)p Fh(\025)2152 2542 y Fe(0)2180 2583 y Fl(])p Fh(:)1376 b Fl(21\))150 2825 y(The)34 b(other)f(inclusion)h(follo)m(ws)f(similarly)-8 b(.)p 1856 2835 60 80 v 349 2945 a(\(Of)34 b(course,)g(the)g(previous)f (lemma)g(is)g(trivial)f(in)h(the)h(case)f(where)i Fh(\025)p Fl(,)e Fh(\025)3112 2909 y Fe(0)3173 2945 y Fl(are)g(in)m(tegers.\))150 3124 y Fm(Lemma)47 b(7)p Fl(.)66 b Fd(If)42 b Fh(\025)h Fd(is)f(an)g(inte)-5 b(ger)42 b(then)g Fh(\025)1814 3088 y Fe(0)1884 3124 y Fd(is)g(also)g(an)h(inte)-5 b(ger.)65 b(F)-8 b(urthermor)j(e,)44 b Fh(\025)f Fd(and)f Fh(\025)3722 3088 y Fe(0)150 3244 y Fd(have)35 b(the)h(same)f(prime)h(factors.)150 3424 y Fl(Pro)s(of:)i(Assume)24 b Fh(\025)g Fl(is)e(an)i(in)m(teger)f (and)g(recall)g Fh(\026)p Fl([\()p Fh(v)t(;)17 b(k)s Fl(\)])k(is)i(de\014ned)i(as)e Fh(\025)2833 3387 y Fe(\000)p Fj(k)2945 3424 y Fh(r)s(v)j Fl(where)e Fh(r)i Fl(is)d(an)g(in-)150 3543 y(teger)28 b(v)m(ector)h(with)e(relativ)m(ely)g(prime)i(en)m (tries.)43 b(Therefore)30 b(there)f(are)f(in)m(tegers)h Fh(v)3244 3558 y Fk(1)3289 3543 y Fh(;)17 b(v)3382 3558 y Fk(2)3426 3543 y Fh(;)g(:)g(:)g(:)d(;)j(v)3696 3558 y Fj(n)150 3663 y Fl(suc)m(h)35 b(that)1732 3706 y Fj(n)1685 3735 y Fi(X)1692 3948 y Fj(i)p Fk(=1)1845 3830 y Fh(v)1893 3845 y Fj(i)1927 3830 y Fh(r)1972 3845 y Fj(i)2033 3830 y Fl(=)28 b(1)p Fh(:)1396 b Fl(22\))150 4109 y(De\014ne)46 b Fh(v)i Fl(to)c(b)s(e)h(the)g(column)g(v)m(ector)g Fh(\025)p Fl(\()p Fh(v)1845 4124 y Fk(1)1889 4109 y Fh(;)17 b(v)1982 4124 y Fk(2)2027 4109 y Fh(;)g(:)g(:)g(:)d(;)j(v)2297 4124 y Fj(n)2351 4109 y Fl(\))2390 4073 y Fk(t)2426 4109 y Fl(.)77 b(Then)46 b Fh(\026)p Fl([\()p Fh(v)t(;)17 b Fl(1\)])44 b(=)j(1)e(and)g(the)150 4229 y(n)m(um)m(b)s(er)35 b(1)e(is)g(in)g(the)h(image)f(of)g Fh(\026)p Fl(.)349 4348 y(F)-8 b(rom)48 b(the)g(previous)g(lemma)g(it)g(is)f(clear)h(that) g Fh(\025)2330 4312 y Fe(0)2406 4348 y Fl(m)m(ust)h(also)e(b)s(e)h(an)h (in)m(teger)f(\(it)f(is)150 4468 y(a)c(P)m(erron)h(eigen)m(v)-6 b(alue)44 b(whic)m(h)g(is)f(either)g(an)h(in)m(teger)f(or)g (irrational\).)72 b(Th)m(us)45 b(for)e(the)h(same)150 4588 y(reason)34 b(1)f(is)g(in)g(the)h(image)f(of)g Fh(\026)1393 4551 y Fe(0)1421 4588 y Fl(.)349 4707 y(Supp)s(ose)i Fh(y)c Ff(2)d Fh(H)1005 4671 y Fe(0)1066 4707 y Fl(with)33 b Fh(\026)1353 4671 y Fe(0)1381 4707 y Fl(\()p Fh(y)t Fl(\))27 b(=)h(1.)44 b(Let)33 b([\()p Fh(u;)17 b(k)s Fl(\)])26 b(=)i Fh(h)2423 4671 y Fe(\000)p Fk(1)2531 4707 y Fl(\()p Fh(y)t Fl(\).)42 b(Then)1306 4929 y Fh(\026)p Fl([\()p Fh(u;)17 b(k)24 b Fl(+)f(1\)])p 1306 4974 521 4 v 1392 5065 a Fh(\026)p Fl([\()p Fh(u;)17 b(k)s Fl(\)])1867 4997 y(=)1984 4929 y Fh(\026)2044 4893 y Fe(0)2072 4929 y Fh(h)p Fl([\()p Fh(u;)g(k)24 b Fl(+)f(1\)])p 1984 4974 607 4 v 2178 5065 a Fh(\026)2238 5036 y Fe(0)2266 5065 y Fl(\()p Fh(y)t Fl(\))1773 5184 y(1)p 1769 5229 59 4 v 1769 5320 a Fh(\025)1867 5252 y Fl(=)28 b Fh(\026)2032 5210 y Fe(0)2060 5252 y Fh(h)p Fl([\()p Fh(u;)17 b(k)24 b Fl(+)f(1\)])p Fh(:)3612 5112 y Fl(23\))1900 5519 y(28)p eop %%Page: 29 30 29 29 bop 150 100 a Fl(This)33 b(implies)h(1)p Fh(=\025)27 b Ff(2)h Fg(Z)p Fl([1)p Fh(=\025)1247 63 y Fe(0)1275 100 y Fl(].)44 b(Similarly)-8 b(,)31 b(1)p Fh(=\025)1976 63 y Fe(0)2031 100 y Ff(2)d Fg(Z)p Fl([1)p Fh(=\025)p Fl(].)44 b(Therefore,)34 b(as)f(subsets)i(of)e Fg(R)p Fl(,)1576 339 y Fg(Z)p Fl([1)p Fh(=\025)p Fl(])27 b(=)i Fg(Z)p Fl([1)p Fh(=\025)2241 298 y Fe(0)2268 339 y Fl(])p Fh(:)1288 b Fl(24\))150 578 y(This)33 b(is)h(the)f(case)h(if)f(and)h (only)f(if)g Fh(\025)g Fl(and)h Fh(\025)1798 542 y Fe(0)1859 578 y Fl(ha)m(v)m(e)g(the)g(same)f(prime)h(factors.)p 3172 588 60 80 v 349 757 a(Com)m(bining)40 b(these)f(results)h(with)f (Theorems)h(3)f(and)g(4)g(then)h(yields)f(our)g(main)g(result,)150 877 y(Theorem)34 b(1.)150 1116 y Fm(8.)51 b(Conclusion.)349 1236 y Fl(W)-8 b(e)31 b(add)g(here)h(some)f(remarks)g(of)g(a)f(general) h(nature.)43 b(The)32 b(basic)f(ob)6 b(jects)31 b(of)g(the)g(ab)s(o)m (v)m(e)150 1356 y(study)39 b(are)g(hierarc)m(hical)g(tilings)f(of)h (the)g(Euclidean)g(plane,)h(whic)m(h)g(are)f(mainly)e(of)i(in)m(terest) 150 1475 y(for)g(their)h(un)m(usual)g(symmetry)f(\(geometric\))g(prop)s (erties.)62 b(Our)40 b(analysis)f(is)g(an)g(outgro)m(wth)150 1595 y(of)f(a)f(traditional)g(dynamical)g(analysis,)h(in)f(whic)m(h)i (a)e(space)i(of)f(tilings)f(is)g(equipp)s(ed)i(with)e(a)150 1714 y(top)s(ological)31 b(and)j(Borel)f(structure)h(together)f(with)g (the)h(action)f(of)g(Euclidean)h(motions,)f(the)150 1834 y(latter)f(main)m(taining)f(con)m(tact)i(with)f(geometry)-8 b(.)43 b(It)32 b(is)g(notable)h(that)f(w)m(e)g(ha)m(v)m(e)h(b)s(een)g (ignoring)150 1953 y(the)43 b(dynamical)f(action,)i(obtaining)e(top)s (ological)e(in)m(v)-6 b(arian)m(ts)43 b(of)f(tiling)g(spaces.)72 b(This)43 b(is)f(a)150 2073 y(recen)m(t)36 b(dev)m(elopmen)m(t,)h(but)e (not)g(new)h(to)e(this)h(pap)s(er,)h(app)s(earing)f(as)h(it)e(has)h(in) h(noncomm)m(u-)150 2192 y(tativ)m(e)f(approac)m(hes.)53 b(Ho)m(w)m(ev)m(er)36 b(a)f(distinctiv)m(e)h(feature)g(of)g(this)g(w)m (ork)f(is)h(that)f(our)h(in)m(v)-6 b(arian)m(t)150 2312 y(is)38 b(only)g(sensible)h(in)f(the)h(top)f(\()1380 2287 y(\024)1369 2312 y(Cec)m(h\))h(cohomology)-8 b(,)37 b(whic)m(h)i(suggests)g(to)f(us)h(that)f(it)f(migh)m(t)150 2432 y(not)h(b)s(e)g(essen)m(tially)g(top)s(ological;)g(more)g(sp)s (eci\014cally)-8 b(,)38 b(it)g(migh)m(t)f(b)s(e)i(in)m(v)-6 b(arian)m(t)37 b(under)j(maps)150 2551 y(more)33 b(general)h(than)f (homeomorphisms.)150 2791 y Fm(Ac)m(kno)m(wledgemen)m(ts.)19 b Fl(W)-8 b(e)23 b(gratefully)e(ac)m(kno)m(wledge)i(useful)h (discussions)f(with)f(Mik)m(e)h(Bo)m(yle,)150 2910 y(Gary)33 b(Hamric)m(k,)f(Ian)h(Putnam,)g(Mik)m(e)h(Starbird)f(and)h(Bob)f (Williams.)1900 5519 y(29)p eop %%Page: 30 31 30 30 bop 150 100 a Fm(References)150 399 y Fl([AnP])41 b(J.)h(Anderson)i(and)e(I.)g(Putnam,)i(T)-8 b(op)s(ological)40 b(in)m(v)-6 b(arian)m(ts)43 b(for)f(substitution)h(tilings)150 519 y(and)34 b(their)f(asso)s(ciated)g(C)1128 483 y Fk(star)1267 519 y Fl(-algebras,)g Fd(Er)-5 b(g.)34 b(Thy.)i(Dyn.)g(Syst.)d Fm(18)h Fl(\(1998\),)d(509-537.)150 698 y([BJV])i(M.)h(Barge)f(and)i (B.)e(Diamond,)h(A)g(complete)g(in)m(v)-6 b(arian)m(t)34 b(for)g(the)h(top)s(ology)d(of)i(substi-)150 818 y(tution)f(tiling)f (spaces,)i(preprin)m(t.)150 997 y([BoT])g(R.)h(Bott)f(and)i(L.)f(T)-8 b(u,)35 b Fd(Di\013er)-5 b(ential)37 b(forms)g(in)g(algebr)-5 b(aic)36 b(top)-5 b(olo)g(gy)p Fl(,)36 b(Graduate)f(T)-8 b(exts)150 1117 y(in)33 b(Mathematics,)g Fm(82)p Fl(,)h(Springer-V)-8 b(erlag,)33 b(New)g(Y)-8 b(ork,)32 b(1982.)150 1296 y([Con])h(A.)f (Connes,)i Fd(Nonc)-5 b(ommutative)36 b(Ge)-5 b(ometry)p Fl(,)34 b(Academic)f(Press,)h(San)g(Diego,)e(1994.)150 1476 y([Ell])h(G.)i(Elliott,)d(On)k(the)f(classi\014cation)f(of)h (inductiv)m(e)g(limits)f(of)h(sequences)i(of)d(semisimple)150 1596 y(\014nite)g(dimensional)f(algebras,)g Fd(J.)j(A)m(lgebr)-5 b(a)33 b Fm(38)h Fl(\(1976\),)d(29-44.)150 1775 y([Go)s(o])52 b(C.)g(Go)s(o)s(dman-Strauss,)59 b(Matc)m(hing)53 b(rules)h(and)f (substitution)g(tilings,)k Fd(A)m(nnals)e(of)150 1895 y(Math.)p Fl(,)33 b Fm(147)h Fl(\(1998\),)d(181-223.)150 2074 y([GPS])48 b(T.)g(Giordano,)k(I.)c(Putnam)h(and)f(C.)g(Sk)-6 b(au,)53 b(T)-8 b(op)s(ological)46 b(orbit)i(equiv)-6 b(alence)49 b(and)150 2194 y(C)222 2158 y Fe(\003)268 2194 y Fl(-crossed)34 b(pro)s(ducts,)g Fd(J.)i(r)-5 b(eine)35 b(angew.)g(Math.)44 b Fm(469)34 b Fl(\(1995\),)d(51-111.)150 2373 y([HPS])36 b(R.)i(Herman,)g(I.)f(Putnam)g(and)h(C.)e(Sk)-6 b(au,)39 b(Ordered)g(Bratteli)d(diagrams,)h(dimension)150 2493 y(groups,)d(and)f(top)s(ological)f(dynamics,)h Fd(Intern.)i(J.)h (Math.)44 b Fm(3)34 b Fl(\(1992\),)d(827-864.)150 2672 y([Ke1])k(J.)g(Kellendonk,)h(Non-comm)m(utativ)m(e)f(geometry)g(of)h (tilings)e(and)i(gap)f(lab)s(elling,)f Fd(R)-5 b(ev.)150 2792 y(Math.)35 b(Phys.)p Fl(,)e Fm(7)g Fl(\(1995\),)f(1133-1180.)150 2972 y([Ke2])j(J.)g(Kellendonk,)g(T)-8 b(op)s(ological)33 b(equiv)-6 b(alence)36 b(of)f(tilings,)f Fd(J.)j(Math.)51 b(Phys.)d Fm(38)36 b Fl(\(1997\),)150 3091 y(1823-1842.)150 3271 y([Ke3])f(J.)h(Kellendonk,)g(The)g(lo)s(cal)e(structure)j(of)f (tilings)e(and)i(their)g(in)m(teger)f(group)h(of)g(coin-)150 3390 y(v)-6 b(arian)m(ts,)33 b Fd(Commun.)k(Math.)e(Phys.)44 b Fm(187)34 b Fl(\(1997\),)d(115-157.)150 3570 y([Ke4])i(J.)g (Kellendonk,)h(On)g(K)1316 3585 y Fk(0)1361 3570 y Fl(-groups)g(for)g (substitution)f(tilings,)f(preprin)m(t.)150 3749 y([KeP])h(J.)g (Kellendonk)h(and)g(I.)f(Putnam,)g(Tilings,)f(C)2197 3713 y Fe(\003)2243 3749 y Fl(-algebras)h(and)h(K-theory)-8 b(,)33 b(preprin)m(t.)150 3929 y([P)m(aT])g(W.)g(P)m(arry)g(and)h(S.)f (T)-8 b(uncel,)34 b Fd(Classi\014c)-5 b(ation)35 b(Pr)-5 b(oblems)36 b(in)f(Er)-5 b(go)g(dic)34 b(The)-5 b(ory)p Fl(,)34 b(Lecture)150 4048 y(Note)f(Series,)h Fm(67)p Fl(,)f(Cam)m(bridge)h(Univ)m(ersit)m(y)e(Press,)i(Cam)m(bridge,)f (1982.)150 4228 y([Put])d(I.)h(Putnam,)g(The)g(ordered)h(K-theory)g(of) f(C)2067 4192 y Fe(\003)2113 4228 y Fl(-algebras)g(asso)s(ciated)g (with)g(substitution)150 4347 y(tilings,)h(preprin)m(t.)150 4527 y([Ra1])41 b(C.)g(Radin,)j(The)e(pin)m(wheel)g(tilings)f(of)g(the) h(plane,)i Fd(A)m(nnals)g(of)e(Math.)69 b Fm(139)43 b Fl(\(1994\),)150 4647 y(661-702.)150 4826 y([Ra2])38 b(C.)h(Radin,)h Fd(Miles)h(of)f(Tiles)p Fl(,)g(Studen)m(t)h(Math.)61 b(Lib.)g Fm(1)p Fl(,)40 b(Amer.)f(Math.)g(So)s(c.,)h(Pro)m(vi-)150 4946 y(dence,)34 b(1999.)150 5125 y([RaS])26 b(C.)h(Radin)g(and)g(L.)g (Sadun,)i(An)d(algebraic)h(in)m(v)-6 b(arian)m(t)26 b(for)h (substitution)g(tiling)f(systems,)150 5245 y Fd(Ge)-5 b(ometriae)35 b(De)-5 b(dic)g(ata)32 b Fm(73)i Fl(\(1998\),)e(21-37.) 1900 5519 y(30)p eop %%Page: 31 32 31 31 bop 150 100 a Fl([RaW])42 b(C.)g(Radin)g(and)h(M.)f(W)-8 b(ol\013,)44 b(Space)f(tilings)e(and)i(lo)s(cal)e(isomorphism,)k Fd(Ge)-5 b(ometriae)150 219 y(De)g(dic)g(ata)32 b Fl(42)h(\(1992\),)f (355-360.)150 399 y([Sol])c(B.)h(Solom)m(y)m(ak,)f(Nonp)s(erio)s(dicit) m(y)h(implies)g(unique)g(comp)s(osition)g(for)g(self-similar)h(trans-) 150 518 y(lationally)h(\014nite)j(tilings,)e Fd(Discr)-5 b(ete)34 b(Comput.)j(Ge)-5 b(om.)44 b Fm(20)34 b Fl(\(1998\),)d (265-279.)150 698 y([SuW])45 b(D.)g(Sulliv)-6 b(an)45 b(and)g(R.F.)f(Williams,)j(The)e(homology)f(of)h(attractors,)i Fd(T)-8 b(op)j(olo)g(gy)45 b Fm(15)150 817 y Fl(\(1976\),)32 b(259-262.)150 1537 y(Nic)m(holas)d(Ormes,)h(Mathematics)g(Departmen)m (t,)g(Univ)m(ersit)m(y)e(of)i(T)-8 b(exas,)29 b(Austin,)g(TX)58 b(78712)150 1656 y Fd(Email)36 b(addr)-5 b(ess)p Fl(:)43 b Fa(ormes@math.utexas.edu)150 1836 y Fl(Charles)35 b(Radin,)h (Mathematics)g(Departmen)m(t,)g(Univ)m(ersit)m(y)e(of)i(T)-8 b(exas,)35 b(Austin,)g(TX)70 b(78712)150 1956 y Fd(Email)36 b(addr)-5 b(ess)p Fl(:)43 b Fa(radin@math.utexas.edu)150 2135 y Fl(Lorenzo)33 b(Sadun,)h(Mathematics)f(Departmen)m(t,)g(Univ)m (ersit)m(y)f(of)h(T)-8 b(exas,)32 b(Austin,)g(TX)65 b(78712)150 2255 y Fd(Email)36 b(addr)-5 b(ess:)44 b Fa(sadun@math.utexas.edu)1900 5519 y Fl(31)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0007281740629--