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b(alen)n(t)32 b(to)h(the)g(fol-)456 2278 y(lo)n(wing)28 b(prop)r(ert)n(y:)41 b(for)29 b(all)h Fq(n)c Fr(2)h Ft(N)40 b Fv(and)30 b(eac)n(h)f Fq(\036;)14 b(\036)2078 2248 y Fp(0)2129 2278 y Fr(2)27 b Ft(Z)2272 2248 y Fo(\(1)p Fs(;n)p Fo(\))2446 2278 y Fr(P)7 b Fv({admissible)29 b(there)g(exists)h(a)456 2390 y Fr(P)7 b Fv({admissible)17 b(arra)n(y)g Fq( )26 b Fr(2)e Ft(Z)1384 2346 y Fo(\()p Fs(m;n)p Fo(\))1384 2399 y Fs(p)1573 2390 y Fv(suc)n(h)19 b(that)h Fq( )s Fr(j)2004 2405 y Fp(f)p Fo(1)p Fp(g\002f)p Fo(1)p Fs(;:::)o(;n)p Fp(g)2425 2390 y Fv(=)i Fq(\036)e Fv(and)f Fq( )s Fr(j)2814 2405 y Fp(f)p Fs(m)p Fp(g\002f)p Fo(1)p Fs(;:::)o(;n)p Fp(g)3261 2390 y Fv(=)k Fq(\036)3398 2359 y Fp(0)3421 2390 y Fv(.)456 2571 y(2.2.)40 b Fy(T)-8 b(op)s(ological)35 b(En)m(trop)m(y.)42 b Fv(It)32 b(can)f(b)r(e)h(sho)n(wn,)g(follo)n(wing)e(a)h(standard)g (subadditivit)n(y)456 2670 y(argumen)n(t,)26 b(that)i(the)g(limit)456 2880 y(\(3\))650 b(lim)1185 2935 y Fs(k)q Fp(!1)1378 2816 y Fv(log)14 b(#)1582 2749 y Fm(\010)1630 2816 y Fq(\036)24 b Fr(2)f Ft(Z)1843 2786 y Fk(n)1884 2795 y Fi(k)1843 2837 y Fs(p)1942 2816 y Fv(:)51 b Fq(\036)28 b Fv(is)f Fr(P)7 b Fv({admissible)2657 2749 y Fm(\011)p 1378 2861 1328 4 v 1791 2937 a Fy(n)1844 2949 y Fs(k)1885 2937 y Fv(\(1\))18 b Fr(\002)g Fy(n)2145 2949 y Fs(k)2186 2937 y Fv(\(2\))456 3080 y(exists)44 b(and)h(has)f(the)h(same)f(v)-5 b(alue)45 b(for)f(all)h(sequence)f Fr(f)p Fy(n)2396 3092 y Fs(k)2437 3080 y Fr(g)2479 3049 y Fp(1)2479 3103 y Fs(k)q Fo(=1)2655 3080 y Fr(\032)51 b Ft(N)40 b Fr(\002)29 b Ft(N)t Fv(,)56 b(suc)n(h)44 b(that)456 3179 y(min\()p Fy(n)679 3191 y Fs(k)720 3179 y Fv(\(1\))p Fq(;)14 b Fy(n)916 3191 y Fs(k)957 3179 y Fv(\(2\)\))36 b Fr(!)h(1)e Fv(when)h Fq(k)i Fr(!)f(1)p Fv(.)60 b(F)-7 b(or)35 b Fq(X)2187 3191 y Fp(P)2278 3179 y Fv(fully)h(extensible,)h(this)f (limit)g(coin-)456 3279 y(cides)28 b(with)h Fn(the)h(top)l(olo)l(gic)l (al)j(entr)l(opy)e(of)g Fv(\000)g Fn(on)f Fq(X)2058 3291 y Fp(P)2114 3279 y Fv(,)f(whic)n(h)f(w)n(e)g(denote)h Fq(h)p Fv(\()p Fq(X)2945 3291 y Fp(P)3001 3279 y Fv(\))g(\(see)f([4)o (])h(for)456 3378 y(details\).)44 b(The)30 b(top)r(ological)e(en)n (trop)n(y)h(measures)g(the)h(exp)r(onen)n(tial)g(gro)n(wth)f(of)h (distinguish-)456 3478 y(able)d(\000{orbits.)456 3679 y(All)e(the)g(notions)f(in)n(tro)r(duced)h(in)g(this)g(subsection)f (can)h(b)r(e)g(de\014ned)g(in)g(a)g(similar)f(w)n(a)n(y)f(in)i(an)n(y) 456 3779 y(dimension.)38 b(W)-7 b(e)28 b(will)h(assume)e(that)i(these)f (notions)f(are)g(w)n(ell)h(kno)n(wn)g(for)f(Mark)n(o)n(v)f(shifts)j(of) 456 3879 y(dimension)e(one.)456 4060 y(2.3.)40 b Fy(T)-8 b(ransition)36 b(Matrices.)41 b Fv(F)-7 b(or)31 b(eac)n(h)g Fq(n)e Fr(\025)g Fv(1,)j(let)g Fq(M)2339 4072 y Fs(n)2413 4060 y Fv(:)e Ft(Z)2527 4017 y Fo(\(1)p Fs(;n)p Fo(\))2527 4070 y Fs(p)2692 4060 y Fr(\002)20 b Ft(Z)2839 4017 y Fo(\(1)p Fs(;n)p Fo(\))2839 4070 y Fs(p)3012 4060 y Fr(!)30 b(f)p Fv(0)p Fq(;)14 b Fv(1)p Fr(g)29 b Fv(b)r(e)456 4159 y(suc)n(h)e(that)456 4339 y(\(4\))531 b Fq(M)1174 4351 y Fs(n)1218 4339 y Fv(\()p Fq(\036;)14 b(\036)1385 4305 y Fp(0)1410 4339 y Fv(\))23 b(=)1553 4222 y Fm(\032)1657 4288 y Fv(1)110 b(if)28 b([)p Fq(\036)g(\036)2034 4258 y Fp(0)2058 4288 y Fv(])g(is)f Fr(P)e(\000)18 b Fv(admissible,)1657 4388 y(0)110 b(otherwise,)456 4564 y(where)30 b([)p Fq(\036)i(\036)852 4534 y Fp(0)876 4564 y Fv(])d Fr(2)h Ft(Z)1074 4521 y Fo(\(2)p Fs(;n)p Fo(\))1074 4573 y Fs(p)1249 4564 y Fv(denotes)h(the)h (concatenation)e(of)h Fq(\036)h Fv(and)f Fq(\036)2627 4534 y Fp(0)2650 4564 y Fv(.)48 b(In)32 b(particular,)e Fq(M)3320 4576 y Fo(1)3389 4564 y Fv(is)456 4663 y(the)e Fq(p)18 b Fr(\002)g Fq(p)p Fv({matrix)26 b(all)h(whose)g(en)n(tries)g (are)g(one.)456 4814 y(Since)j(\()p Fq(M)788 4826 y Fs(n)833 4814 y Fv(\)\))897 4784 y Fk(m)p Fp(\000)p Fo(1)1049 4814 y Fv(\()p Fq(\036;)14 b(\036)p Fv(\))32 b(is)e(the)h(sum)f(o)n(v)n (er)f(all)h(the)g(c)n(hains)g Fq(\036)e Fr(7!)f Fq(\036)2625 4826 y Fo(1)2691 4814 y Fr(7!)g(\001)14 b(\001)g(\001)28 b(7!)f Fq(\036)3085 4826 y Fs(m)p Fp(\000)p Fo(2)3261 4814 y Fr(!)h Fq(\036)3421 4784 y Fp(0)456 4913 y Fv(of)f(allo)n(w)n (ed)f(transitions)h(starting)g(with)h Fq(\036)g Fv(and)f(ending)h(b)n (y)f Fq(\036)2426 4883 y Fp(0)2450 4913 y Fv(,)h(it)g(follo)n(ws)e (that)456 5065 y(\(5\))523 b(#)1168 4998 y Fm(\010)1217 5065 y Fq(\036)23 b Fr(2)h Ft(Z)1429 5031 y Fk(m)1429 5086 y Fs(p)1513 5065 y Fv(:)51 b Fq(\036)28 b Fv(is)f Fr(P)7 b Fv({admissible)2228 4998 y Fm(\011)2299 5065 y Fv(=)40 b(I)-50 b(1)2425 5029 y Fp(y)2425 5086 y Fs(n)2470 5065 y Fq(M)2560 5031 y Fs(m)p Fp(\000)p Fo(1)2551 5086 y Fs(n)2725 5065 y Fv(I)g(1)2746 5077 y Fs(n)2792 5065 y Fq(;)456 5216 y Fv(where)43 b(I)-50 b(1)734 5228 y Fs(n)807 5216 y Fv(is)27 b(the)h(all{ones)e(column)i(v)n(ector)e(of)i (dimension)f Fq(p)2416 5185 y Fs(n)2461 5216 y Fv(.)p eop %%Page: 3 3 3 2 bop 752 251 a Fo(EIGENV)-7 b(ALUE)29 b(EQUA)-5 b(TION)28 b(F)n(OR)i(THE)f(ENTR)n(OPY)g(OF)g(2{D)g(MARK)n(O)n(V)g(SHIFTS)263 b(3)456 454 y Fv(F)-7 b(or)27 b(eac)n(h)f Fy(a)e Fr(2)f Ft(Z)1001 411 y Fo(\(2)p Fs(;)o Fo(1\))1001 464 y Fs(p)1164 454 y Fv(and)28 b(eac)n(h)e Fq(n)d Fr(\025)g Fv(1)k(let)h Fq(M)1952 424 y Fk(a)1943 474 y Fs(n)2015 454 y Fv(:)c Ft(Z)2123 411 y Fo(\(1)p Fs(;n)p Fo(\))2123 464 y Fs(p)2285 454 y Fr(\002)18 b Ft(Z)2430 411 y Fo(\(1)p Fs(;)o(n)p Fo(\))2430 464 y Fs(p)2597 454 y Fr(!)23 b(f)p Fv(0)p Fq(;)14 b Fv(1)p Fr(g)25 b Fv(b)r(e)j(suc)n(h)g(that)456 647 y(\(6\))258 b Fq(M)910 613 y Fk(a)901 668 y Fs(n)950 647 y Fv(\()p Fq(\036;)14 b(\036)1117 613 y Fp(0)1142 647 y Fv(\))23 b(=)1285 530 y Fm(\032)1388 596 y Fq(M)1478 566 y Fk(a)1469 617 y Fo(1)1519 596 y Fv(\()p Fq(\036)p Fv(\()p Fq(n)p Fv(\))p Fq(;)14 b(\036)1800 566 y Fp(0)1824 596 y Fv(\()p Fq(n)p Fv(\)\))112 b(if)28 b([)p Fq(\036)g(\036)2307 566 y Fp(0)2331 596 y Fv(])g(is)f Fr(P)e(\000)18 b Fv(admissible,)1659 696 y(0)381 b(otherwise.)456 879 y(The)27 b(incidence)h(sub{matrices)e Fq(M)1575 849 y Fk(a)1566 900 y Fs(n)1643 879 y Fv(satisfy)i(the)g (relations)1337 1026 y Fq(M)1427 992 y Fk(a)1418 1047 y Fs(n)p Fo(+1)1630 1026 y Fv(=)83 b([)p Fq(M)1891 992 y Fk(a)1882 1047 y Fo(1)1931 1026 y Fv(\()p Fy(b)p Fv(\))19 b Fr(\002)f Fq(M)2240 992 y Fk(b)2231 1047 y Fs(n)2286 1026 y Fv(])2309 1059 y Fk(b)p Fp(2)p Fl(Z)2440 1029 y Fh(\(2)p Fi(;)p Fh(1\))2440 1068 y Fi(p)456 1026 y Fv(\(7\))1337 1172 y Fq(M)1418 1184 y Fs(n)p Fo(+1)1630 1172 y Fv(=)83 b([)p Fq(M)1891 1138 y Fk(a)1882 1193 y Fs(n)1931 1172 y Fv(])1954 1205 y Fk(a)p Fp(2)p Fl(Z)2080 1175 y Fh(\(2)p Fi(;)p Fh(1\))2080 1214 y Fi(p)2203 1172 y Fq(:)-1770 b Fv(\(8\))456 1342 y(This)32 b(notation)h(indicates)g (the)g(decomp)r(osition)f(of)h(a)g(matrix)f(of)h(dimension)g Fq(p)3030 1312 y Fs(n)p Fo(+1)3192 1342 y Fv(in)n(to)f Fq(p)3407 1312 y Fo(2)456 1441 y Fv(sub{matrices)26 b(of)h(dimension)h Fq(p)1484 1411 y Fs(n)1529 1441 y Fv(.)456 1596 y(Belo)n(w)h(w)n(e)i (will)f(use)h(the)g Fn(the)i(Kr)l(oneker)g(pr)l(o)l(duct)p Fv(.)46 b(F)-7 b(or)30 b Fq(A)e Fv(:)h Ft(Z)2477 1553 y Fo(\(2)p Fs(;)p Fo(1\))2477 1606 y Fs(p)2641 1596 y Fr(!)f Ft(N)41 b Fv(and)31 b Fq(B)h Fv(:)c Ft(Z)3215 1553 y Fo(\(1)p Fs(;n)p Fo(\))3215 1606 y Fs(p)3380 1596 y Fr(\002)456 1712 y Ft(Z)517 1669 y Fo(\(1)p Fs(;n)p Fo(\))517 1722 y Fs(p)684 1712 y Fr(!)23 b Ft(N)38 b Fv(it)28 b(is)f(the)h(matrix)f Fq(A)19 b Fr(\012)f Fq(B)27 b Fv(:)c Ft(Z)1819 1669 y Fo(\(1)p Fs(;)o(n)p Fo(+1\))1819 1722 y Fs(p)2065 1712 y Fr(\002)18 b Ft(Z)2209 1669 y Fo(\(1)p Fs(;n)p Fo(+1\))2209 1722 y Fs(p)2460 1712 y Fr(!)23 b Ft(N)38 b Fv(suc)n(h)27 b(that)456 1905 y(\(9\))565 b(\()p Fq(A)19 b Fr(\012)f Fq(B)t Fv(\))1436 1788 y Fm(\022)q(\024)1586 1855 y Fq(a)1583 1954 y(\036)1674 1788 y Fm(\025)1732 1905 y Fq(;)1769 1788 y Fm(\024)1857 1855 y Fq(a)1901 1825 y Fp(0)1854 1954 y Fq(\036)1903 1924 y Fp(0)1968 1788 y Fm(\025\023)2096 1905 y Fv(=)23 b Fq(A)p Fv(\()p Fq(a;)14 b(a)2403 1871 y Fp(0)2426 1905 y Fv(\))p Fq(B)t Fv(\()p Fq(\036;)g(\036)2692 1871 y Fp(0)2717 1905 y Fv(\))p Fq(;)456 2123 y Fv(for)27 b(all)g Fq(a;)14 b(a)823 2093 y Fp(0)869 2123 y Fr(2)23 b Ft(Z)1009 2135 y Fs(p)1069 2123 y Fv(and)k Fq(\036;)14 b(\036)1365 2093 y Fp(0)1412 2123 y Fr(2)24 b Ft(Z)1552 2080 y Fo(\(1)p Fs(;n)p Fo(\))1552 2133 y Fs(p)1696 2123 y Fv(.)456 2310 y(As)38 b(w)n(e)f(p)r(oin)n(ted)i (out)f(b)r(efore,)i(the)f(top)r(ological)d(en)n(trop)n(y)h Fq(h)p Fv(\()p Fq(X)2523 2322 y Fp(P)2579 2310 y Fv(\))h(is)g(indep)r (enden)n(t)h(of)f(the)456 2410 y(sequence)c Fr(f)p Fy(n)902 2422 y Fs(k)942 2410 y Fr(g)984 2379 y Fp(1)984 2433 y Fs(k)q Fo(=1)1143 2410 y Fv(used)h(in)f(the)h(limiting)g(pro)r(cess)e (de\014ning)i(it.)58 b(In)35 b(the)g(case)e(of)i(a)f(fully)456 2509 y(extensible)e(Mark)n(o)n(v)e(shift)i Fq(X)1413 2521 y Fp(P)1469 2509 y Fv(,)i(as)d(a)h(direct)g(consequence)f(of)h (this)h(fact,)g(and)f(taking)g(in)n(to)456 2609 y(accoun)n(t)26 b(equation)h(\(5\),)h(w)n(e)f(obtain)1232 2863 y Fq(h)p Fv(\()p Fq(X)1381 2875 y Fp(P)1437 2863 y Fv(\))c(=)52 b(lim)1580 2913 y Fs(n)p Fp(!1)1781 2807 y Fv(1)p 1777 2844 50 4 v 1777 2920 a Fq(n)1889 2863 y Fv(lim)1851 2913 y Fs(m)p Fp(!1)2066 2777 y Fv(log)2187 2685 y Fm(\020)2253 2777 y Fv(I)-50 b(1)2275 2741 y Fp(y)2309 2777 y Fq(M)2399 2747 y Fs(m)p Fp(\000)p Fo(1)2390 2798 y Fs(n)2564 2777 y Fv(I)g(1)2585 2685 y Fm(\021)p 2066 2844 570 4 v 2314 2920 a Fq(m)2645 2863 y(:)456 3075 y Fv(If)32 b(in)h(addition)f(w)n(e)h (assume)e(that)i Fq(X)1650 3087 y Fp(P)1738 3075 y Fv(is)g(horizon)n (tally)d(transitiv)n(e,)j(then)g(eac)n(h)f(one)g(of)g(the)456 3174 y(matrices)37 b Fq(M)880 3186 y Fs(n)963 3174 y Fv(has)g(to)h(b)r(e)g(irreducible)g(\(see)g([1)o(])h(for)e(details\).) 68 b(Then)38 b(P)n(erron{F)-7 b(rob)r(enius)456 3274 y(Theory)26 b(applies,)h(and)h(w)n(e)f(obtain)456 3452 y(\(10\))848 b Fq(h)p Fv(\()p Fq(X)1601 3464 y Fp(P)1657 3452 y Fv(\))23 b(=)52 b(lim)1800 3502 y Fs(n)p Fp(!1)2001 3396 y Fv(1)p 1997 3433 50 4 v 1997 3509 a Fq(n)2071 3452 y Fv(log)14 b Fq(\032)p Fv(\()p Fq(M)2348 3464 y Fs(n)2393 3452 y Fv(\))p Fq(;)456 3624 y Fv(where)31 b Fq(\032)p Fv(\()p Fq(M)9 b Fv(\))32 b(denotes)f(the)h(sp)r(ectral)f (radius)g(of)h(the)g(matrix)f Fq(M)9 b Fv(.)49 b(F)-7 b(or)31 b(non{negativ)n(e)f(irre-)456 3724 y(ducible)e(matrices)e(it)i (coincides)f(with)h(the)g(maximal)f(eigen)n(v)-5 b(alue.)456 3934 y(2.4.)40 b Fy(Thermo)s(dynamical)23 b(F)-8 b(ormalism.)58 b Fv(Let)22 b Fq(X)29 b Fr(\032)23 b Ft(Z)2322 3904 y Fl(N)2322 3955 y Fs(p)2384 3934 y Fv(b)r(e)e(a)g(one{dimensional)f (Mark)n(o)n(v)456 4037 y(shift,)28 b(and)f Fq(f)32 b Fv(:)23 b Fq(X)29 b Fr(!)24 b Ft(R)33 b Fv(H\177)-42 b(older)27 b(con)n(tin)n(uous.)36 b(It)28 b(is)g(pro)n(v)n(en)e(in)h ([2])h(that)g(the)g(limit)971 4329 y Fq(P)1024 4341 y Fs(X)1088 4329 y Fv(\()p Fq(f)9 b Fv(\))23 b(=)51 b(lim)1312 4379 y Fs(n)p Fp(!1)1514 4273 y Fv(1)p 1510 4310 V 1510 4386 a Fq(n)1583 4329 y Fv(log)1704 4113 y Fm(0)1704 4259 y(B)1704 4309 y(B)1704 4362 y(@)1894 4251 y(X)1848 4438 y Fi(\036)p Fj(2)p Fg(Z)1955 4417 y Fi(n)p Fh(+1)1955 4449 y Fi(p)1787 4495 y(X)s Fh({admissible)2145 4329 y Fv(exp)2286 4163 y Fm(0)2286 4312 y(@)2398 4226 y Fs(n)2359 4251 y Fm(X)2361 4427 y Fs(j)s Fo(=0)2493 4329 y Fq(f)9 b Fv(\()p Fq(\033)2625 4295 y Fs(j)2660 4329 y Fq(x)2707 4341 y Fs(\036)2751 4329 y Fv(\))2783 4163 y Fm(1)2783 4312 y(A)2856 4113 y(1)2856 4259 y(C)2856 4309 y(C)2856 4362 y(A)456 4622 y Fv(exists,)42 b(for)d(an)n(y)g Fq(x)1078 4634 y Fs(\036)1166 4622 y Fr(2)44 b Fq(X)i Fv(suc)n(h)39 b(that)h Fq(x)p Fr(j)1841 4637 y Fp(f)p Fo(0)p Fs(;)p Fo(1)p Fs(;:::)o(;n)p Fp(g)2183 4622 y Fv(=)i Fq(\036)p Fv(.)74 b(The)40 b(quan)n(tit)n(y)f Fq(P)3016 4634 y Fs(X)3080 4622 y Fv(\()p Fq(f)9 b Fv(\))39 b(is)h(the)456 4722 y Fn(top)l(olo)l(gic)l(al)31 b(pr)l(essur)l(e)d(of)h(the)g(p)l (otential)h Fq(f)9 b Fv(.)36 b(Asso)r(ciated)26 b(to)g(the)h(p)r(oten)n (tial)f Fq(f)9 b Fv(,)27 b(there)f(exists)g(a)456 4821 y(unique)19 b Fq(\033)s Fv({in)n(v)-5 b(arian)n(t)17 b(probabilit)n(y)h(measure)g Fq(\026)1929 4833 y Fs(f)1991 4821 y Fv(c)n(haracterized)f(b)n(y)i(the)g(follo)n(wing)f(inequalit)n (y)-7 b(.)456 4921 y(There)27 b(exists)g(a)g(constan)n(t)g Fq(C)i Fr(\025)23 b Fv(1)k(suc)n(h)g(that)1136 5108 y Fq(C)1201 5074 y Fp(\000)p Fo(1)1314 5108 y Fr(\024)1473 5047 y Fq(\026)1523 5059 y Fs(f)1566 5047 y Fr(f)p Fq(x)c Fr(2)g Fq(X)29 b Fv(:)51 b Fq(x)p Fr(j)1998 5062 y Fp(f)p Fo(0)p Fs(;)p Fo(1)p Fs(;:::;n)p Fp(g)2320 5047 y Fv(=)22 b Fq(\036)p Fr(g)p 1411 5089 1149 4 v 1411 5195 a Fv(exp)1552 5103 y Fm(\020)1601 5133 y(P)1689 5153 y Fs(n)1689 5220 y(j)s Fo(=0)1822 5195 y Fq( )s Fv(\()p Fq(\033)1961 5171 y Fs(j)1997 5195 y Fq(x)2044 5207 y Fs(\036)2088 5195 y Fv(\))d Fr(\000)f Fq(nP)2325 5207 y Fs(X)2388 5195 y Fv(\()p Fq( )s Fv(\))2509 5103 y Fm(\021)2592 5108 y Fr(\024)23 b Fq(C)q(;)p eop %%Page: 4 4 4 3 bop 456 251 a Fo(4)1135 b(EDGARDO)23 b(UGALDE)456 450 y Fv(for)i(eac)n(h)h Fq(X)7 b Fv({admissible)25 b Fq(\036)e Fr(2)h Ft(Z)1497 420 y Fs(n)p Fo(+1)1497 471 y Fs(p)1620 450 y Fv(.)37 b(The)26 b(measure)f Fq(\026)2221 462 y Fs(f)2291 450 y Fv(is)h(the)h(Gibbs)f(measure)g(asso)r(ciated)456 553 y(to)h(the)h(p)r(oten)n(tial)g Fq(f)9 b Fv(.)36 b(Tw)n(o)27 b(p)r(oten)n(tials)g Fq(f)36 b Fv(and)28 b Fq(f)2019 523 y Fp(0)2069 553 y Fv(are)f Fn(c)l(ohomolo)l(gous)j Fv(if)e Fq(\026)2852 565 y Fs(f)2918 553 y Fv(=)23 b Fq(\026)3056 565 y Fs(f)3095 548 y Fj(0)3121 553 y Fv(.)456 704 y(A)j(relation)e(b)r(et)n(w)n(een)i(the)g(top)r(ological)e (pressure)h(and)g(the)h(Gibbs)g(measure)f(asso)r(ciated)f(to)i(a)456 804 y(giv)n(en)g(p)r(oten)n(tial,)i(is)f(giv)n(en)g(b)n(y)h(the)g Fn(variational)k(principle)456 956 y Fv(\(11\))637 b Fq(P)1294 968 y Fs(X)1358 956 y Fv(\()p Fq(f)9 b Fv(\))23 b Fr(\025)g Fq(\026)p Fv(\()p Fq(f)9 b Fv(\))18 b(+)g Fq(h)p Fv(\()p Fq(\026)p Fv(\))59 b Fr(8)p Fq(\026)e(\033)s Fv({in)n(v)-5 b(arian)n(t,)456 1108 y(where)40 b Fq(h)p Fv(\()p Fq(\026)p Fv(\))i(denotes)f(the)h(en)n(trop)n(y)e(of)h(the)h (measure)e(\(see)i([2)o(]\).)79 b(F)-7 b(or)40 b Fq(X)48 b Fv(top)r(ologically)456 1207 y(transitiv)n(e,)23 b(i.)i(e.,)f(when)g (its)h(transition)e(matrix)g(is)h(irreducible,)g(the)h(equalit)n(y)e (holds)g(only)h(for)456 1307 y Fq(\026)f Fv(=)f Fq(\026)666 1319 y Fs(f)709 1307 y Fv(.)1339 1526 y(3.)41 b Fu(Entr)n(opy)31 b(as)g(an)h(Eigenv)-7 b(alue)456 1675 y Fv(3.1.)40 b Fy(Eigen)m(v)-5 b(alues')41 b(Recursion.)f Fv(Denote)e(b)n(y)e Ft(R)2135 1632 y Fo(\()p Fs(p)q(;p)p Fo(\))2135 1696 y(+)2323 1675 y Fv(the)h(set)g(of)h(p)r(ositiv)n(e)e(matrices)h(of)456 1798 y(dimension)27 b Fq(p)18 b Fr(\002)g Fq(p)p Fv(,)28 b(and)f(consider)g(the)h(transformation)e Fq(R)e Fv(:)f Ft(R)2465 1755 y Fo(\()p Fs(p;p)p Fo(\))2465 1819 y(+)2638 1798 y Fr(!)g Ft(R)2798 1755 y Fo(\()p Fs(p;p)p Fo(\))2798 1819 y(+)2976 1798 y Fv(suc)n(h)k(that)456 1950 y(\(12\))852 b Fq(R)q(M)31 b Fv(=)23 b([)p Fq(\032)p Fv(\()p Fq(M)1908 1915 y Fk(a)1899 1970 y Fo(1)1949 1950 y Fv(\))18 b Fr(\003)g Fq(M)9 b Fv(])2172 1983 y Fk(a)p Fp(2)p Fl(Z)2298 1952 y Fh(\(2)p Fi(;)p Fh(1\))2298 1992 y Fi(p)2421 1950 y Fq(;)456 2118 y Fv(where)27 b Fr(\003)g Fv(stands)g(for)g(the)h(co)r (ordinate-wise)e(m)n(ultiplication.)456 2270 y(When)i(for)f(eac)n(h)g Fq(n)d Fr(2)g Ft(N)38 b Fv(all)27 b(the)h(transition)g(matrices)f Fq(M)2314 2240 y Fk(a)2305 2290 y Fs(n)2382 2270 y Fv(comm)n(ute,)h (the)g(transformation)456 2369 y(just)g(de\014ned)g(can)f(b)r(e)h(used) g(to)f(recursiv)n(ely)f(compute)h Fq(\032)p Fv(\()p Fq(M)2368 2381 y Fs(n)2413 2369 y Fv(\))h(for)f(eac)n(h)g Fq(n)c Fr(2)h Ft(N)s Fv(.)456 2547 y Fy(Lemma)49 b(1.)i Fn(Assume)45 b(that)h(the)g(matric)l(es)g Fq(M)2046 2517 y Fk(a)2037 2568 y Fs(n)2132 2547 y Fn(c)l(ommute)f(for)i(e)l(ach)f Fq(n)52 b Fr(2)h Ft(N)t Fn(.)92 b(Then,)456 2663 y Fq(\032)p Fv(\()p Fq(M)612 2675 y Fs(n)p Fo(+1)741 2663 y Fv(\))23 b(=)g Fq(\032)p Fv(\()p Fq(R)1023 2633 y Fs(n)1114 2663 y Fn(I)-50 b(1)17 b(I)-50 b(1)1180 2627 y Fp(y)1214 2663 y Fv(\))30 b Fn(and)h Fq(\032)p Fv(\()p Fq(M)1603 2633 y Fk(a)1594 2683 y Fs(n)1643 2663 y Fv(\))24 b(=)e(\()p Fq(R)1882 2633 y Fs(n)1974 2663 y Fn(I)-50 b(1)17 b(I)-50 b(1)2040 2627 y Fp(y)2074 2663 y Fv(\)\()p Fy(a)p Fv(\))31 b Fn(for)g(al)t(l)f Fq(n)23 b Fr(2)h Ft(N)39 b Fn(and)31 b Fy(a)23 b Fr(2)h Ft(Z)3110 2620 y Fo(\(2)p Fs(;)p Fo(1\))3110 2673 y Fs(p)3246 2663 y Fn(.)456 2837 y(Pr)l(o)l(of.)43 b Fv(W)-7 b(e)38 b(use)f(the)h(follo)n(wing)e(fact.)67 b(Supp)r(ose)37 b(that)h Fq(M)48 b Fv(=)39 b([)p Fq(N)2618 2807 y Fk(a)2659 2837 y Fv(])2682 2870 y Fk(a)p Fp(2)p Fl(Z)2808 2840 y Fh(\(2)p Fi(;)p Fh(1\))2808 2879 y Fi(p)2968 2837 y Fv(and)e(that)h(the)456 2958 y(submatrices)26 b Fq(N)989 2928 y Fk(a)1058 2958 y Fv(comm)n(ute,)h(then)h Fq(\032)p Fv(\()p Fq(M)9 b Fv(\))23 b(=)g Fq(\032)p Fv(\([)p Fq(\032)p Fv(\()p Fq(N)2184 2928 y Fk(a)2225 2958 y Fv(\)])2280 2991 y Fk(a)p Fp(2)p Fl(Z)2406 2960 y Fh(\(2)p Fi(;)p Fh(1\))2406 3000 y Fi(p)2529 2958 y Fv(\).)456 3136 y(By)k(h)n(yp)r (othesis)g(the)h Fq(p)1180 3106 y Fo(2)1245 3136 y Fv(submatrices)e(of) i Fq(M)1878 3148 y Fo(2)1942 3136 y Fv(comm)n(ute,)g(then)1294 3289 y Fq(\032)p Fv(\()p Fq(M)1450 3301 y Fo(2)1487 3289 y Fv(\))23 b(=)g Fq(\032)p Fv(\([)p Fq(\032)p Fv(\()p Fq(M)1893 3255 y Fk(a)1884 3310 y Fo(1)1934 3289 y Fv(\)])1989 3322 y Fk(a)p Fp(2)p Fl(Z)2115 3292 y Fh(\(2)p Fi(;)p Fh(1\))2115 3331 y Fi(p)2238 3289 y Fv(\))g(=)g Fq(R)45 b Fv(I)-50 b(1)16 b(I)-50 b(1)2549 3253 y Fp(y)2583 3289 y Fq(:)456 3539 y Fv(Note)26 b(that)h(\()p Fq(R)44 b Fv(I)-50 b(1)16 b(I)-50 b(1)1033 3503 y Fp(y)1067 3539 y Fv(\)\()p Fy(a)p Fv(\))25 b(=)d Fq(\032)p Fv(\()p Fq(M)1486 3509 y Fk(a)1477 3559 y Fo(1)1527 3539 y Fv(\))27 b(for)f(eac)n(h)f Fy(a)f Fr(2)f Ft(Z)2106 3496 y Fo(\(2)p Fs(;)p Fo(1\))2106 3549 y Fs(p)2242 3539 y Fv(.)37 b(Supp)r(ose)26 b(that)h(for)f Fq(n)d Fr(\025)g Fv(2)j(w)n(e)g(ha)n(v)n(e)456 3655 y Fq(\032)p Fv(\()p Fq(M)621 3625 y Fk(a)612 3675 y Fs(n)661 3655 y Fv(\))d(=)g(\()p Fq(R)900 3625 y Fs(n)945 3655 y Fq(U)9 b Fv(\)\()p Fy(a)p Fv(\))29 b(for)e(all)g Fy(a)d Fr(2)f Ft(Z)1634 3612 y Fo(\(2)p Fs(;)o Fo(1\))1634 3665 y Fs(p)1770 3655 y Fv(.)36 b(Since)28 b(the)g(matrices)f Fq(M)2612 3625 y Fk(b)2603 3675 y Fs(n)2685 3655 y Fv(comm)n(ute,)h(w)n (e)f(ha)n(v)n(e)1041 3811 y Fq(\032)p Fv(\()p Fq(M)1206 3777 y Fk(a)1197 3832 y Fs(n)p Fo(+1)1326 3811 y Fv(\))84 b(=)e Fq(\032)p Fv(\()p Fq(M)1754 3777 y Fk(a)1745 3832 y Fo(1)1795 3811 y Fv(\()p Fy(b)p Fv(\))19 b Fr(\002)f Fq(\032)p Fv(\([)p Fq(M)2202 3777 y Fk(b)2193 3832 y Fs(n)2248 3811 y Fv(\)])2303 3844 y Fk(b)p Fp(2)p Fl(Z)2434 3814 y Fh(\(2)p Fi(;)p Fh(1\))2434 3853 y Fi(p)2557 3811 y Fv(\)\))1442 3971 y(=)82 b Fq(\032)p Fv(\()p Fq(M)1754 3937 y Fk(a)1745 3991 y Fo(1)1813 3971 y Fr(\003)18 b Fv(\()p Fq(R)1969 3937 y Fs(n)2059 3971 y Fv(I)-50 b(1)16 b(I)-50 b(1)2119 3935 y Fp(y)2153 3971 y Fv(\)\))24 b(=)e(\()p Fq(R)2424 3937 y Fs(n)p Fo(+1)2598 3971 y Fv(I)-50 b(1)16 b(I)-50 b(1)2658 3935 y Fp(y)2692 3971 y Fv(\)\()p Fy(a)p Fv(\))p Fq(:)456 4139 y Fv(W)-7 b(e)25 b(ha)n(v)n(e)f(pro)n(v)n(ed)f(b) n(y)i(that)g Fq(\032)p Fv(\()p Fq(M)1507 4109 y Fk(a)1498 4159 y Fs(n)1548 4139 y Fv(\))e(=)g(\()p Fq(R)1787 4109 y Fs(n)1874 4139 y Fv(I)-50 b(1)16 b(I)-50 b(1)1934 4103 y Fp(y)1968 4139 y Fv(\)\()p Fy(a)p Fv(\))27 b(for)d(all)h Fq(n)e Fr(2)g Ft(N)35 b Fv(and)25 b Fy(a)f Fr(2)f Ft(Z)2978 4096 y Fo(\(2)p Fs(;)p Fo(1\))2978 4149 y Fs(p)3114 4139 y Fv(.)36 b(Finally)-7 b(,)456 4239 y(since)27 b Fq(M)740 4251 y Fs(n)p Fo(+1)892 4239 y Fv(=)22 b([)p Fq(M)1092 4208 y Fk(a)1083 4259 y Fs(n)1133 4239 y Fv(])1156 4271 y Fk(a)p Fp(2)p Fl(Z)1282 4241 y Fh(\(2)p Fi(;)p Fh(1\))1282 4281 y Fi(p)1432 4239 y Fv(and)28 b(the)g(sub{matrices)e(comm)n(ute,)i (the)g(result)f(follo)n(ws.)3380 4353 y Ff(\003)456 4505 y Fv(Denote)19 b(b)n(y)g(\001)908 4475 y Fo(\()p Fs(p;p)p Fo(\))1075 4505 y Fr(\032)k Ft(R)1217 4462 y Fo(\()p Fs(p;p)p Fo(\))1217 4526 y(+)1386 4505 y Fv(the)d(set)f(all)g Fq(p)r Fr(\002)r Fq(p)e Fv(matrices)h(with)i(p)r(ositiv)n(e)f(en)n (tries)f(and)h(maximal)456 4604 y(eigen)n(v)-5 b(alue)26 b(equals)h(to)h(1,)f(and)g(de\014ne)h(the)g(normalized)f (transformation)456 4791 y(\(13\))582 b Fq(F)35 b Fv(:)23 b(\001)1389 4757 y Fo(\()p Fs(p;p)p Fo(\))1557 4791 y Fr(!)g Fv(\001)1732 4757 y Fo(\()p Fs(p;p)p Fo(\))1876 4791 y Fq(;)69 b(F)12 b(M)32 b Fr(\021)2353 4735 y Fv(1)p 2244 4772 261 4 v 2244 4848 a Fq(\032)p Fv(\()p Fq(R)q(M)9 b Fv(\))2542 4791 y Fq(R)q(M)t(:)456 5039 y Fv(Let)27 b(us)h(remark)e(that)i(for)f(a)g(fully)h(extensible)g(and)f(horizon)n (tally)f(transitiv)n(e)h(Mark)n(o)n(v)e(shift,)1289 5191 y Fq(h)p Fv(\()p Fq(X)1438 5203 y Fp(P)1494 5191 y Fv(\))e(=)52 b(lim)1637 5240 y Fs(n)p Fp(!1)1824 5191 y Fv(log\()p Fq(\032)p Fv(\()p Fq(M)2119 5203 y Fs(n)p Fo(+1)2249 5191 y Fv(\))p Fq(=\032)p Fv(\()p Fq(M)2479 5203 y Fs(n)2523 5191 y Fv(\)\))p Fq(;)p eop %%Page: 5 5 5 4 bop 752 251 a Fo(EIGENV)-7 b(ALUE)29 b(EQUA)-5 b(TION)28 b(F)n(OR)i(THE)f(ENTR)n(OPY)g(OF)g(2{D)g(MARK)n(O)n(V)g(SHIFTS)263 b(5)456 450 y Fv(whenev)n(er)27 b(this)i(limit)h(exists.)40 b(In)29 b(this)g(case,)f(if)i(the)f(sub{matrices)e Fq(M)2729 420 y Fk(a)2720 471 y Fs(n)2798 450 y Fv(comm)n(ute)i(for)f(eac)n(h)456 550 y Fq(n)22 b Fr(2)i Ft(N)t Fv(,)34 b(w)n(e)27 b(ha)n(v)n(e)456 772 y(\(14\))306 b Fq(h)p Fv(\()p Fq(X)1059 784 y Fp(P)1115 772 y Fv(\))23 b(=)52 b(lim)1258 822 y Fs(n)p Fp(!1)1445 772 y Fv(log)1566 631 y Fm( )1642 716 y Fq(\032)p Fv(\()p Fq(R)1781 686 y Fs(n)p Fo(+1)1954 716 y Fv(I)-50 b(1)17 b(I)-50 b(1)2014 680 y Fp(y)2049 716 y Fv(\))p 1642 753 440 4 v 1684 840 a Fq(\032)p Fv(\()p Fq(R)1823 816 y Fs(n)1912 840 y Fv(I)g(1)17 b(I)-50 b(1)1972 804 y Fp(y)2007 840 y Fv(\))2091 631 y Fm(!)2180 772 y Fv(=)52 b(lim)2268 822 y Fs(n)p Fp(!1)2455 772 y Fv(log\()p Fq(R)29 b(F)2751 738 y Fs(n)2840 772 y Fv(I)-50 b(1)16 b(I)-50 b(1)2900 736 y Fp(y)2935 772 y Fv(\))p Fq(:)456 1067 y Fv(Belo)n(w)24 b(w)n(e)h(giv)n(e)g(conditions)g(ensuring)f(that)i(the)g(limit)g Fq(F)2293 1037 y Fs(n)2380 1067 y Fv(I)-50 b(1)17 b(I)-50 b(1)2440 1031 y Fp(y)2500 1067 y Fv(exists.)36 b(F)-7 b(or)25 b(this)h(w)n(e)f(use)g(the)456 1167 y Fn(pr)l(oje)l(ctive)31 b(distanc)l(e)d Fv(in)g(\001)1312 1137 y Fo(\()p Fs(p;p)p Fo(\))1484 1167 y Fv(de\014ned)g(b)n(y)456 1365 y(\(15\))180 b Fq(d)p Fv(\()p Fq(M)t(;)14 b(M)1071 1330 y Fp(0)1094 1365 y Fv(\))24 b(=)67 b(max)1237 1440 y Fk(a)p Fp(2)p Fl(Z)1363 1410 y Fh(\(2)p Fi(;)p Fh(1\))1363 1449 y Fi(p)1496 1248 y Fm(\032)1558 1365 y Fv(log)1679 1248 y Fm(\022)1762 1308 y Fq(M)9 b Fv(\()p Fy(a)p Fv(\))p 1750 1346 224 4 v 1750 1422 a Fq(M)1840 1398 y Fp(0)1863 1422 y Fv(\()p Fy(a)p Fv(\))1984 1248 y Fm(\023\033)2126 1365 y Fr(\000)71 b Fv(min)2209 1440 y Fk(a)p Fp(2)p Fl(Z)2335 1410 y Fh(\(2)p Fi(;)p Fh(1\))2335 1449 y Fi(p)2467 1248 y Fm(\032)2530 1365 y Fv(log)2651 1248 y Fm(\022)2734 1308 y Fq(M)9 b Fv(\()p Fy(a)p Fv(\))p 2722 1346 V 2722 1422 a Fq(M)2812 1398 y Fp(0)2835 1422 y Fv(\()p Fy(a)p Fv(\))2956 1248 y Fm(\023\033)3093 1365 y Fq(:)456 1656 y Fv(If)26 b(the)g(limit)h Fq(M)956 1668 y Fs(f)1022 1656 y Fv(=)22 b(lim)1225 1668 y Fs(n)p Fp(!1)1416 1656 y Fq(F)1481 1626 y Fs(n)1569 1656 y Fv(I)-50 b(1)16 b(I)-50 b(1)1629 1620 y Fp(y)1690 1656 y Fv(exists,)26 b(then)g(it)h(has)e(to)h(b)r(e)g(a)g(\014xed)g(p)r (oin)n(t)g(for)g Fq(F)12 b Fv(,)26 b(i.)g(e.,)456 1756 y Fq(F)12 b(M)602 1768 y Fs(f)673 1756 y Fv(=)28 b Fq(M)847 1768 y Fs(f)889 1756 y Fv(.)47 b(Th)n(us,)31 b(under)g(suitable)g (conditions,)g(the)g(computation)g(of)g(the)g(top)r(ological)456 1856 y(en)n(trop)n(y)26 b(reduces)h(to)g(the)h(computation)g(of)f(a)g (\014xed)h(p)r(oin)n(t.)456 2008 y(The)f(next)h(theorem)f(giv)n(es)g (su\016cien)n(t)g(conditions)g(ensuring)g(this)h(con)n(v)n(ergence.)456 2252 y Fy(Lemma)34 b(2.)43 b Fn(L)l(et)33 b Fq(R)e Fv(:)f Ft(R)1265 2209 y Fo(\()p Fs(p;p)p Fo(\))1265 2273 y(+)1445 2252 y Fr(!)f Ft(R)1611 2209 y Fo(\()p Fs(p)q(;p)p Fo(\))1611 2273 y(+)1795 2252 y Fn(and)34 b Fq(F)42 b Fv(:)30 b(\001)2177 2222 y Fo(\()p Fs(p;p)p Fo(\))2351 2252 y Fr(!)g Fv(\001)2533 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b(=)f(log)14 b Fq(M)9 b Fv(\()p Fq(x)p Fv(\(0\))p Fq(;)14 b(x)p Fv(\(1\)\).)456 4065 y(The)27 b(pro)5 b(jectiv)n(e)27 b(distance)g(extends)h(to)f(a)g(pseudo{metric)g (in)h Ft(R)2509 4022 y Fo(\()p Fs(p;p)p Fo(\))2509 4086 y(+)2686 4065 y Fv(satisfying)1229 4224 y Fq(d)p Fv(\()p Fq(M)t(;)14 b(M)1516 4190 y Fp(0)1539 4224 y Fv(\))24 b(=)e Fq(d)p Fv(\()p Fq(\025M)t(;)14 b(\025)1975 4190 y Fp(0)2000 4224 y Fq(M)2090 4190 y Fp(0)2113 4224 y Fv(\))55 b Fr(8)27 b Fq(\025;)14 b(\025)2407 4190 y Fp(0)2454 4224 y Fr(2)24 b Ft(R)2587 4190 y Fo(+)2648 4224 y Fq(;)456 4377 y Fv(implying)41 b(that)h Fq(d)p Fv(\()p Fq(F)54 b(M)t(;)14 b(F)53 b(M)1507 4347 y Fp(0)1530 4377 y Fv(\))47 b(=)f Fq(d)p Fv(\()p Fq(R)c(M)t(;)14 b(R)43 b(M)2218 4347 y Fp(0)2240 4377 y Fv(\).)79 b(F)-7 b(rom)41 b(this)h(and)g(the)g (fact)f(that)456 4476 y(\()p Fq(R)28 b(M)9 b Fv(\)\()p Fy(a)p Fv(\))24 b(=)f(exp)13 b(\()q Fq(P)1149 4488 y Fs(X)1208 4472 y Fe(a)1247 4476 y Fv(\()p Fq(f)1320 4488 y Fs(M)1394 4476 y Fv(\)\),)28 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Ft(Z)933 411 y Fo(\(2)p Fs(;)p Fo(1\))933 464 y Fs(p)1097 454 y Fv(b)r(e)28 b(suc)n(h)f(that)881 611 y(min)828 686 y Fk(a)p Fp(2)p Fl(Z)954 656 y Fh(\(2)p Fi(;)p Fh(1\))954 696 y Fi(p)1086 611 y Fr(f)p Fq(P)1181 623 y Fs(X)1240 607 y Fe(a)1279 611 y Fv(\()p Fq(f)1352 623 y Fs(M)1426 611 y Fv(\))18 b Fr(\000)g Fq(P)1612 623 y Fs(X)1671 607 y Fe(a)1711 611 y Fv(\()p Fq(f)1784 623 y Fs(M)1853 607 y Fj(0)1880 611 y Fv(\))p Fr(g)82 b Fv(=)h Fq(P)2237 628 y Fs(X)2296 611 y Fe(b)8 b Fv(\()p Fq(f)2412 623 y Fs(M)2486 611 y Fv(\))19 b Fr(\000)f Fq(P)2673 628 y Fs(X)2732 611 y Fe(b)8 b Fv(\()p Fq(f)2848 623 y Fs(M)2917 607 y Fj(0)2944 611 y Fv(\))p Fq(;)873 807 y Fv(max)828 883 y Fk(a)p Fp(2)p Fl(Z)954 852 y Fh(\(2)p Fi(;)p Fh(1\))954 892 y Fi(p)1086 807 y Fr(f)p Fq(P)1181 819 y Fs(X)1240 803 y Fe(a)1279 807 y Fv(\()p Fq(f)1352 819 y Fs(M)1426 807 y Fv(\))18 b Fr(\000)g Fq(P)1612 819 y Fs(X)1671 803 y Fe(a)1711 807 y Fv(\()p Fq(f)1784 819 y Fs(M)1853 803 y Fj(0)1880 807 y Fv(\))p Fr(g)82 b 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1310 y Fj(0)2126 1314 y Fv(\()p Fy(a)p Fv(\))p Fr(g)h(\000)71 b Fv(min)2380 1389 y Fk(a)p Fp(2)p Fl(Z)2506 1359 y Fh(\(2)p Fi(;)p Fh(1\))2506 1398 y Fi(p)2625 1314 y Fr(f)p Fq(f)2708 1326 y Fs(M)2780 1314 y Fv(\()p Fy(a)p Fv(\))20 b Fr(\000)e Fq(f)3034 1326 y Fs(M)3103 1310 y Fj(0)3130 1314 y Fv(\()p Fy(a)p Fv(\))p Fr(g)1229 1514 y Fv(=)82 b Fq(d)p Fv(\()p Fq(M)t(;)14 b(M)1663 1480 y Fp(0)1686 1514 y Fv(\))p Fq(:)456 1729 y Fv(The)27 b(\014rst)h(inequalit)n(y)f(is)h(strict)g (unless)f Fq(f)1771 1741 y Fs(M)1872 1729 y Fv(and)h Fq(f)2075 1741 y Fs(M)2144 1725 y Fj(0)2199 1729 y Fv(are)e (cohomologous)f(in)j(b)r(oth)h Fq(X)3238 1699 y Fk(b)3311 1729 y Fv(and)456 1832 y Fq(X)532 1802 y Fk(b)574 1777 y Fj(0)599 1832 y Fv(.)35 b(The)22 b(second)f(inequalit)n(y)g(is)g (strict)h(unless)f(\()p Fq(f)2064 1844 y Fs(M)2144 1832 y Fr(\000)6 b Fq(f)2256 1844 y Fs(M)2325 1828 y Fj(0)2352 1832 y Fv(\))22 b(is)g(constan)n(t)f(on)g Fq(X)2998 1802 y Fk(b)3065 1832 y Fv(and)g(equals)456 1939 y(its)27 b(maximal)g(v)-5 b(alue,)28 b(while)g(on)f Fq(X)1556 1909 y Fk(b)1598 1884 y Fj(0)1651 1939 y Fv(is)h(constan)n(t)f(and)g (equals)g(its)h(minimal)g(v)-5 b(alue.)456 2100 y(Hyp)r(othesis)43 b Fq(a)p Fv(\))h(implies)f(that)h(there)f(are)g(p)r(oin)n(ts)g Fq(x)50 b Fr(2)g Fq(X)2438 2070 y Fk(b)2527 2100 y Fv(and)43 b Fq(y)52 b Fr(2)e Fq(X)2978 2070 y Fk(b)3020 2045 y Fj(0)3089 2100 y Fv(suc)n(h)43 b(that)456 2199 y(\()p Fq(x)p Fv(\(1\))p Fq(;)14 b(x)p Fv(\(2\)\))26 b(=)d(\()p Fq(y)s Fv(\(1\))p Fq(;)14 b(y)s Fv(\(2\)\),)29 b(therefore)f(\()p Fq(f)1853 2211 y Fs(M)1945 2199 y Fr(\000)19 b Fq(f)2070 2211 y Fs(M)2139 2195 y Fj(0)2166 2199 y Fv(\)\()p Fq(x)p Fv(\))25 b(=)f(\()p Fq(f)2496 2211 y Fs(M)2589 2199 y Fr(\000)18 b Fq(f)2713 2211 y Fs(M)2782 2195 y Fj(0)2809 2199 y Fv(\)\()p Fq(y)s Fv(\).)40 b(Hence,)29 b(ev)n(en)456 2307 y(if)d Fq(f)571 2319 y Fs(M)670 2307 y Fv(and)g Fq(f)871 2319 y Fs(M)940 2302 y Fj(0)992 2307 y Fv(are)f(cohomologous)e (in)j(b)r(oth)g Fq(X)2023 2276 y Fk(b)2094 2307 y Fv(and)g Fq(X)2330 2276 y Fk(b)2372 2251 y Fj(0)2397 2307 y Fv(,)g(the)g(second) g(inequalit)n(y)f(ab)r(o)n(v)n(e)456 2406 y(has)i(to)g(b)r(e)h(strict)g (for)f(all)g Fq(M)32 b Fr(6)p Fv(=)22 b Fq(M)1568 2376 y Fp(0)1591 2406 y Fv(.)456 2621 y(By)33 b(h)n(yp)r(othesis)h Fq(b)p Fv(\),)i Fq(F)46 b Fv(has)34 b(a)f(\014x)h(p)r(oin)n(t)h Fq(M)1893 2633 y Fs(f)1969 2621 y Fr(2)g Fv(\001)2128 2591 y Fo(\()p Fs(p;p)p Fo(\))2272 2621 y Fv(.)56 b(F)-7 b(or)34 b(arbitrary)e Fq(M)43 b Fr(2)34 b Fv(\001)3149 2591 y Fo(\()p Fs(p;p)p Fo(\))3293 2621 y Fv(,)i(let)456 2727 y Fq(C)515 2739 y Fs(M)631 2727 y Fr(\021)43 b(f)p Fq(M)871 2697 y Fp(0)936 2727 y Fr(2)g Fv(\001)1103 2697 y Fo(\()p Fs(p;p)p Fo(\))1290 2727 y Fv(:)82 b Fq(d)p Fv(\()p Fq(M)1560 2697 y Fp(0)1583 2727 y Fq(;)14 b(M)1701 2739 y Fs(f)1744 2727 y Fv(\))43 b Fr(\024)f Fq(d)p Fv(\()p Fq(M)t(;)14 b(M)2204 2739 y Fs(f)2247 2727 y Fv(\))p Fr(g)p Fq(:)39 b Fv(This)g(set)h(is)f(compact)g(and)g Fq(F)12 b Fv({)456 2827 y(in)n(v)-5 b(arian)n(t.)35 b(Indeed,)1013 2984 y Fq(d)p Fv(\()p Fq(F)40 b(M)1271 2950 y Fp(0)1294 2984 y Fq(;)14 b(M)1412 2996 y Fs(f)1455 2984 y Fv(\))23 b Fq(<)g(d)p Fv(\()p Fq(M)1763 2950 y Fp(0)1786 2984 y Fq(;)14 b(M)1904 2996 y Fs(f)1946 2984 y Fv(\))24 b Fr(\024)e Fq(d)p Fv(\()p Fq(M)t(;)14 b(M)2367 2996 y Fs(f)2410 2984 y Fv(\))28 b Fr(8)p Fq(M)2607 2950 y Fp(0)2652 2984 y Fr(2)c Fq(C)2790 2996 y Fs(M)2864 2984 y Fq(:)456 3202 y Fv(Since)j Fr(f)p Fq(d)p Fv(\()p Fq(F)854 3172 y Fs(k)923 3202 y Fq(M)t(;)14 b(M)1126 3214 y Fs(f)1168 3202 y Fv(\))p Fr(g)1242 3172 y Fp(1)1242 3225 y Fs(k)q Fo(=0)1394 3202 y Fv(is)28 b(a)f(decreasing)f(sequence,)h(then)h Fq(r)e Fv(=)c(lim)2771 3214 y Fs(k)q Fp(!1)2958 3202 y Fq(d)p Fv(\()p Fq(F)3098 3172 y Fs(k)3167 3202 y Fq(M)t(;)14 b(M)3370 3214 y Fs(f)3412 3202 y Fv(\))456 3304 y(exists.)34 b(F)-7 b(urthermore,)20 b(eac)n(h)g(one)f(of)i(the)f(matrices)f Fq(F)2150 3274 y Fs(k)2211 3304 y Fq(M)29 b Fv(b)r(elongs)20 b(to)g(the)g(compact)g(set)g Fq(C)3347 3316 y Fs(M)3421 3304 y Fv(,)456 3404 y(then)30 b(there)h(exists)f(an)g(accum)n(ulation) f(p)r(oin)n(t)h Fq(M)2034 3374 y Fp(\003)2100 3404 y Fv(=)d(lim)2307 3416 y Fs(n)p Fp(!1)2499 3404 y Fq(F)2564 3374 y Fs(k)2599 3382 y Fi(n)2674 3404 y Fq(M)9 b Fv(,)31 b(whic)n(h)f(necessarily)456 3504 y(satis\014es)c Fq(r)g Fv(=)d Fq(d)p Fv(\()p Fq(M)1077 3473 y Fp(\003)1115 3504 y Fq(;)14 b(M)1233 3516 y Fs(f)1276 3504 y Fv(\).)37 b(If)28 b Fq(r)e(>)c Fv(0,)28 b(i.)f(e.,)h(if)g Fq(M)2044 3473 y Fp(\003)2105 3504 y Fr(6)p Fv(=)23 b Fq(M)2274 3516 y Fs(f)2316 3504 y Fv(,)28 b(then)1092 3668 y Fq(r)e(>)c(d)p Fv(\()p Fq(F)41 b(M)1501 3634 y Fp(\003)1538 3668 y Fq(;)14 b(M)1656 3680 y Fs(f)1699 3668 y Fv(\))23 b(=)52 b(lim)1842 3718 y Fs(n)p Fp(!1)2029 3668 y Fq(d)p Fv(\()p Fq(F)2169 3634 y Fs(k)2204 3642 y Fi(n)2245 3634 y Fo(+1)2361 3668 y Fq(M)t(;)14 b(M)2564 3680 y Fs(f)2607 3668 y Fv(\))23 b(=)g Fq(r)n(;)456 3850 y Fv(whic)n(h)18 b(is)g(imp)r(ossible.)34 b(Therefore)17 b Fq(r)26 b Fv(=)c(0,)e(and)f(the)f(lemma)h(follo)n(ws.) 2957 b Ff(\003)456 4056 y Fv(3.2.)40 b Fy(Classes)33 b(and)i(Comm)m(utativit)m(y.)k Fv(Giv)n(en)30 b Fq(X)2161 4068 y Fp(P)2216 4056 y Fv(,)h(the)f(set)f Ft(Z)2608 4013 y Fo(\(2)p Fs(;)p Fo(1\))2608 4066 y Fs(p)2774 4056 y Fv(can)g(b)r(e)h(partitioned)456 4156 y(in)i(classes)g Fq(C)886 4168 y Fo(1)923 4156 y Fq(;)14 b(C)1019 4168 y Fo(2)1057 4156 y Fq(;)g(:)g(:)g(:)f(;)h(C)1300 4168 y Fs(K)1365 4156 y Fv(,)33 b(de\014ned)g(according)e(to)h(the)h(equiv) -5 b(alen)n(t)33 b(relation)e Fy(a)h Fr(\030)f Fy(a)3274 4126 y Fp(0)3361 4156 y Fr(,)456 4258 y Fq(M)546 4228 y Fk(a)537 4279 y Fo(1)609 4258 y Fv(=)23 b Fq(M)787 4228 y Fk(a)824 4203 y Fj(0)778 4279 y Fo(1)849 4258 y Fv(.)456 4416 y(The)k(classes)g(are)f(preserv)n(ed)g(along)g(the)i (recursion)f(\(7\).)37 b(Indeed,)27 b(if)i Fy(a)23 b Fr(\030)g Fy(a)2858 4386 y Fp(0)2909 4416 y Fv(then)1293 4577 y Fq(M)1383 4543 y Fk(a)1374 4598 y Fs(n)p Fo(+1)1585 4577 y Fv(=)1733 4510 y Fm(\000)1771 4577 y Fq(M)1861 4543 y Fk(a)1852 4598 y Fo(1)1901 4577 y Fv(\()p Fy(b)p Fv(\))c Fr(\002)f Fq(M)2210 4543 y Fk(b)2201 4598 y Fs(n)2256 4510 y Fm(\001)2294 4610 y Fk(b)p Fp(2)p Fl(Z)2425 4580 y Fh(\(2)p Fi(;)p Fh(1\))2425 4619 y Fi(p)1585 4756 y Fv(=)1733 4664 y Fm(\020)1783 4756 y Fq(M)1873 4722 y Fk(a)1910 4697 y Fj(0)1864 4777 y Fo(1)1935 4756 y Fv(\()p Fy(b)p Fv(\))h Fr(\002)f Fq(M)2244 4722 y Fk(b)2235 4777 y Fs(n)2290 4664 y Fm(\021)2340 4814 y Fk(b)p Fp(2)p Fl(Z)2471 4784 y Fh(\(2)p Fi(;)p Fh(1\))2471 4823 y Fi(p)1585 4947 y Fv(=)83 b Fq(M)1823 4913 y Fk(a)1860 4888 y Fj(0)1814 4968 y Fs(n)1886 4947 y Fq(;)456 5104 y Fv(for)40 b(all)g Fy(n)46 b Fr(2)f Ft(N)t Fv(.)83 b(Hence,)44 b(to)d(eac)n(h)f(class)g(w) n(e)g(can)h(asso)r(ciate)e Fn(r)l(epr)l(esentative)j(tr)l(ansition)456 5216 y(matric)l(es)28 b Fq(M)878 5172 y Fo([)p Fs(k)q Fo(])869 5225 y Fs(n)979 5216 y Fr(\021)22 b Fq(M)1156 5185 y Fk(a)1147 5236 y Fs(n)1224 5216 y Fv(for)28 b(an)n(y)e Fy(a)e Fr(2)f Fq(C)1715 5228 y Fs(k)1756 5216 y Fv(.)p eop %%Page: 7 7 7 6 bop 752 251 a Fo(EIGENV)-7 b(ALUE)29 b(EQUA)-5 b(TION)28 b(F)n(OR)i(THE)f(ENTR)n(OPY)g(OF)g(2{D)g(MARK)n(O)n(V)g(SHIFTS)263 b(7)456 450 y Fv(The)29 b(represen)n(tativ)n(e)d(transition)j(matrices) f(inherit)h(the)g(recurrence)e(relation)h(giv)n(en)g(in)h(\(7\),)456 550 y(whic)n(h)e(w)n(e)h(rewrite)f(b)n(y)h(using)f(the)i(Kronek)n(er)c (pro)r(duct.)38 b(F)-7 b(or)28 b(this)g(let)g(us)g(de\014ne)g(the)h Fn(linking)456 661 y(matric)l(es)f Fq(L)845 631 y Fo([)p Fs(k)q(;`)p Fo(])993 661 y Fv(:)23 b Ft(Z)1101 618 y Fo(\(2)p Fs(;)o Fo(1\))1101 671 y Fs(p)1260 661 y Fr(!)g(f)p Fv(0)p Fq(;)14 b Fv(1)p Fr(g)25 b Fv(for)i Fq(k)s(;)14 b(`)23 b Fr(2)g(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(g)p Fv(,)27 b(suc)n(h)g(that)1550 820 y Fq(L)1607 786 y Fo([)p Fs(k)q(;`)p Fo(])1733 820 y Fv(\()p Fy(b)p Fv(\))c(=)g Fq(M)2051 777 y Fo([)p Fs(k)q Fo(])2042 842 y(1)2148 820 y Fr(\003)34 b Fv(I)-50 b(1)2246 832 y Fs(C)2294 841 y Fi(`)2327 820 y Fq(;)456 981 y Fv(where)38 b(I)-50 b(1)729 993 y Fs(C)777 1002 y Fi(`)833 981 y Fv(:)23 b Ft(Z)940 938 y Fo(\(2)p Fs(;)p Fo(1\))940 991 y Fs(p)1099 981 y Fr(!)g(f)p Fv(0)p Fq(;)14 b Fv(1)p Fr(g)21 b Fv(denotes)h(the)i (c)n(haracteristic)c(function)k(of)e(the)h(class)f Fq(C)3167 993 y Fs(`)3199 981 y Fv(.)36 b(With)456 1081 y(this)27 b(w)n(e)h(ha)n(v)n(e)1430 1284 y Fq(M)1511 1296 y Fs(n)p Fo(+1)1723 1284 y Fv(=)1901 1180 y Fs(K)1871 1205 y Fm(X)1870 1384 y Fs(k)q Fo(=1)2021 1284 y Fv(I)-50 b(1)2043 1296 y Fs(C)2091 1305 y Fi(k)2150 1284 y Fr(\012)18 b Fq(M)2323 1250 y Fo([)p Fs(k)q Fo(])2314 1305 y Fs(n)456 1284 y Fv(\(17\))1430 1577 y Fq(M)1520 1534 y Fo([)p Fs(k)q Fo(])1511 1599 y Fs(n)p Fo(+1)1723 1577 y Fv(=)1900 1474 y Fs(K)1870 1498 y Fm(X)1874 1677 y Fs(`)p Fo(=1)2004 1577 y Fq(L)2061 1543 y Fo([)p Fs(k)q(;`)p Fo(])2205 1577 y Fr(\012)g Fq(M)2378 1543 y Fo([)p Fs(`)p Fo(])2369 1598 y Fs(n)2447 1577 y Fq(;)-2014 b Fv(\(18\))456 1799 y(for)27 b(all)g Fq(k)f Fr(2)d(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(g)27 b Fv(and)g Fq(n)c Fr(2)h Ft(N)t Fv(.)456 1921 y Fy(Lemma)29 b(3.)41 b Fn(Supp)l(ose)30 b(that)g(al)t(l)h(the)e Fq(K)24 b Fr(\002)18 b Fq(K)36 b Fn(linking)30 b(matric)l(es)g(c)l(ommute.)38 b(Then,)1008 2067 y Fq(M)1098 2032 y Fo([)p Fs(k)q Fo(])1089 2087 y Fs(n)1176 2067 y Fq(M)1266 2032 y Fo([)p Fs(`)p Fo(])1257 2087 y Fs(n)1358 2067 y Fv(=)23 b Fq(M)1536 2032 y Fo([)p Fs(`)p Fo(])1527 2087 y Fs(n)1605 2067 y Fq(M)1695 2032 y Fo([)p Fs(k)q Fo(])1686 2087 y Fs(n)1832 2067 y Fr(8)29 b Fq(k)s(;)14 b(`)23 b Fr(2)g(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(g)p Fq(;)43 b(n)22 b Fr(2)i Ft(N)t Fq(;)456 2226 y Fn(whenever)30 b Fq(M)907 2183 y Fo([)p Fs(k)q Fo(])898 2248 y(1)985 2226 y Fq(M)1075 2183 y Fo([)p Fs(`)p Fo(])1066 2248 y(1)1168 2226 y Fv(=)22 b Fq(M)1345 2183 y Fo([)p Fs(`)p Fo(])1336 2248 y(1)1414 2226 y Fq(M)1504 2183 y Fo([)p Fs(k)q Fo(])1495 2248 y(1)1612 2226 y Fn(for)31 b(al)t(l)f Fq(k)s(;)14 b(`)23 b Fr(2)g(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(g)456 2387 y Fn(Pr)l(o)l(of.)43 b Fv(This)33 b(follo)n(ws)g(b)n(y) g(induction,)j(taking)d(in)n(to)g(accoun)n(t)g(that)g(the)h(Kronek)n (er)e(pro)r(duct)456 2487 y(satis\014es)1308 2595 y(\()p Fq(A)19 b Fr(\012)f Fq(B)t Fv(\)\()p Fq(A)1697 2561 y Fp(0)1740 2595 y Fr(\012)g Fq(B)1890 2561 y Fp(0)1913 2595 y Fv(\))23 b(=)g(\()p Fq(AA)2212 2561 y Fp(0)2236 2595 y Fv(\))c Fr(\012)f Fv(\()p Fq(B)t(B)2536 2561 y Fp(0)2560 2595 y Fv(\))456 2721 y(for)27 b(all)g(matrices)g Fq(A;)14 b(A)1192 2691 y Fp(0)1243 2721 y Fv(and)28 b Fq(B)t(;)14 b(B)1576 2691 y Fp(0)1627 2721 y Fv(of)27 b(compatible)h(dimension.)3380 2821 y Ff(\003)456 2987 y Fv(3.3.)40 b Fy(Main)32 b(Result.)40 b Fv(W)-7 b(e)29 b(will)f(consider)f(a)h(Mark)n(o)n(v)e(shift)i Fq(X)2467 2999 y Fp(P)2523 2987 y Fv(,)h(whic)n(h)f(is)g(fully)g(extensible)456 3098 y(and)33 b(horizon)n(tally)e(transitiv)n(e.)53 b(Let)34 b Fq(M)1749 3055 y Fo([1])1740 3120 y(1)1823 3098 y Fq(;)14 b(:)g(:)g(:)g(;)g(M)2098 3055 y Fo([)p Fs(K)t Fo(])2089 3120 y(1)2232 3098 y Fv(b)r(e)34 b(the)g(represen)n(tativ)n(e)d (matrices)i(of)456 3215 y(the)c(transition)f(submatrices)g Fr(f)p Fq(M)1571 3185 y Fk(a)1562 3236 y Fo(1)1635 3215 y Fv(:)54 b Fy(a)26 b Fr(2)f Ft(Z)1925 3172 y Fo(\(2)p Fs(;)p Fo(1\))1925 3225 y Fs(p)2061 3215 y Fr(g)p Fv(,)k(and)f Fr(f)p Fq(L)2416 3185 y Fo([)p Fs(k)q(;`)p Fo(])2566 3215 y Fv(:)h Fq(;)14 b(`)25 b Fr(2)g(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(gg)27 b Fv(the)456 3315 y(linking)g(matrices)g(relating)f(them.)456 3458 y(If)d(the)h(represen)n(tativ)n(e)d(matrices)i(comm)n(ute,)h(as)e(w)n (ell)h(as)g(the)g(linking)h(matrices,)f(then,)i(under)456 3558 y(some)i(additional)g(conditions,)g(the)h(en)n(trop)n(y)e(is)i (the)g(solution)f(of)g(an)h(eigen)n(v)-5 b(alue)26 b(problem.)456 3679 y Fy(Theorem)j(1)h Fv(\(En)n(trop)n(y)24 b(as)i(an)g(Eigen)n(v)-5 b(alue\))p Fy(.)39 b Fn(Supp)l(ose)29 b(that)f(the)h(r)l(epr)l (esentative)g(matric)l(es)456 3779 y(c)l(ommute,)44 b(as)e(wel)t(l)h (as)f(the)g(linking)h(matric)l(es.)75 b(Supp)l(ose)42 b(also)h(that)f(the)g(r)l(epr)l(esentative)456 3890 y(matric)l(es)30 b(ar)l(e)h(irr)l(e)l(ducible)h(and)f(such)f(that)h Fq(M)1948 3847 y Fo([)p Fs(k)q Fo(])1939 3912 y(1)2045 3890 y Fr(\003)18 b Fq(M)2195 3847 y Fo([)p Fs(`)p Fo(])2186 3912 y(1)2289 3890 y Fr(6)p Fv(=)23 b Fy(0)p Fn(,)31 b(for)g(al)t(l)h Fq(k)s(;)14 b(`)23 b Fr(2)i(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(g)p Fn(.)456 3990 y(Then)30 b(if)h(the)f(system)f(of)i(e)l(quations)1127 4209 y Fq(\025)19 b Fr(\002)f Fq(x)1324 4221 y Fs(k)1388 4209 y Fv(=)23 b Fq(\032)1533 4067 y Fm( )1628 4105 y Fs(K)1598 4130 y Fm(X)1603 4309 y Fs(`)p Fo(=1)1749 4209 y Fn(I)-50 b(1)1773 4221 y Fs(C)1821 4230 y Fi(`)1873 4209 y Fr(\003)18 b Fq(M)2023 4166 y Fo([)p Fs(k)q Fo(])2014 4231 y(1)2101 4067 y Fm(!)2180 4209 y Fq(;)c(k)26 b Fv(=)d(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)q(;)456 4431 y Fn(has)30 b(a)g(p)l(ositive)h(solution,)g(then)e Fq(\025)i Fn(is)f(uniquely)g (determine)l(d)g(and)g Fq(\025)24 b Fv(=)e(exp\()p Fq(h)p Fv(\()p Fq(X)3051 4443 y Fp(P)3108 4431 y Fv(\)\))p Fn(.)456 4593 y(Pr)l(o)l(of.)43 b Fv(W)-7 b(e)34 b(need)g Fq(X)1134 4605 y Fp(P)1224 4593 y Fv(to)f(b)r(e)h(fully)h(extensible)f(and)f (horizon)n(tally)f(transitiv)n(e)h(in)h(order)f(to)456 4693 y(relate)27 b(the)i(en)n(trop)n(y)e(to)h(the)g(maximal)g(eigen)n (v)-5 b(alue)27 b(of)h(the)h(transition)e(matrices)h Fq(M)3149 4705 y Fs(n)3194 4693 y Fv(.)38 b(Since)456 4792 y(the)21 b(represen)n(tativ)n(e)f(matrices)g(comm)n(ute,)i(as)f(w) n(ell)g(as)g(the)g(linking)g(matrices,)h(then)g(Lemma)f(3)456 4892 y(holds,)30 b(and)h(with)g(this)f(\(14\))g(applies.)45 b(On)30 b(the)h(other)f(hand,)h(the)g(represen)n(tativ)n(e)d(matrices) 456 5008 y(are)f(primitiv)n(e)i(and)f Fq(M)1204 4965 y Fo([)p Fs(k)q Fo(])1195 5030 y(1)1301 5008 y Fr(\003)19 b Fq(M)1452 4965 y Fo([)p Fs(`)p Fo(])1443 5030 y(1)1546 5008 y Fr(6)p Fv(=)25 b Fy(0)j Fv(for)g(eac)n(h)g Fq(k)s(;)14 b(`)p Fv(,)28 b(then)i(Lemma)e(\(2\))h(holds.)40 b(Therefore,)456 5107 y(whenev)n(er)1311 5216 y Fq(\025)19 b Fr(\002)f Fq(M)9 b Fv(\()p Fy(a)p Fv(\))24 b(=)f Fq(\032)p Fv(\()p Fq(M)k Fr(\003)18 b Fq(M)2106 5172 y Fk(a)p Fo(\))2097 5238 y(1)2172 5216 y Fv(\))p Fq(;)c Fy(a)24 b Fr(2)f Ft(Z)2450 5181 y Fo(\()p Fs(p;p)p Fo(\))2450 5236 y Fs(p)p eop %%Page: 8 8 8 7 bop 456 251 a Fo(8)1135 b(EDGARDO)23 b(UGALDE)456 454 y Fv(has)28 b(a)g(p)r(ositiv)n(e)g(solution)g Fq(M)34 b Fr(2)25 b Ft(R)1547 411 y Fo(\()p Fs(p;p)p Fo(\))1547 474 y(+)1697 454 y Fv(,)k(necessarily)e Fq(\025)f Fv(=)e(exp\()p Fq(h)p Fv(\()p Fq(X)2635 466 y Fs(p)2674 454 y Fq(p)p Fv(\)\).)40 b(Assuming)29 b(this)g(is)456 553 y(true,)c(since)f(the)h (righ)n(t)f(hand)h(side)f(of)h(the)g(previous)e(equation)i(remains)e (unc)n(hanged)h(inside)h(a)456 653 y(class)i Fq(C)710 665 y Fs(k)751 653 y Fv(,)h(then)h(there)f(is)g(a)g(constan)n(t)f Fq(x)1741 665 y Fs(k)1807 653 y Fq(>)c Fv(0)28 b(suc)n(h)g(that)g Fq(M)9 b Fv(\()p Fy(a)p Fv(\))25 b(=)e Fq(x)2693 665 y Fs(k)2763 653 y Fv(for)28 b Fy(a)c Fr(2)g Fq(C)3099 665 y Fs(k)3141 653 y Fv(.)38 b(Hence,)456 764 y(w)n(e)27 b(ma)n(y)g(write)g Fq(M)32 b Fv(=)1170 701 y Fm(P)1258 722 y Fs(K)1258 788 y(k)q Fo(=1)1397 764 y Fq(x)1444 776 y Fs(k)1502 764 y Fv(I)-50 b(1)1523 776 y Fs(C)1571 785 y Fi(k)1611 764 y Fv(,)28 b(and)g(the)g(previous)e(equation)h(b)r (ecomes)887 994 y(exp\()p Fq(h)p Fv(\()p Fq(X)1195 1006 y Fs(p)1233 994 y Fq(p)p Fv(\)\))19 b Fr(\002)f Fq(x)1488 1006 y Fs(k)1552 994 y Fv(=)23 b Fq(\032)1697 852 y Fm( )1792 891 y Fs(K)1762 916 y Fm(X)1766 1094 y Fs(`)p Fo(=1)1896 994 y Fq(x)1943 1006 y Fs(`)1992 994 y Fv(I)-50 b(1)2014 1006 y Fs(C)2062 1015 y Fi(`)2113 994 y Fr(\003)18 b Fq(M)2263 951 y Fo([)p Fs(k)q Fo(])2254 1017 y(1)2341 852 y Fm(!)2421 994 y Fq(;)c(k)25 b Fv(=)e(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)q(;)456 1214 y Fv(and)27 b(so)g(the)h(theorem)f(follo)n(ws.)3380 1314 y Ff(\003)456 1485 y Fv(3.4.)40 b Fy(Conditioned)i(Pro)s(ducts.)g Fv(In)c(this)g (subsection)f(w)n(e)h(exhibit)g(a)g(class)f(of)h(fully)g(ex-)456 1585 y(tensible)31 b(and)g(horizon)n(tally)e(transitiv)n(e)h(Mark)n(o)n (v)f(shifts,)j(for)e(whic)n(h)h(all)g(the)g(h)n(yp)r(otheses)g(of)456 1685 y(Theorem)24 b(1)i(hold.)36 b(Therefore,)25 b(their)g(en)n(trop)n (y)g(is)g(an)h(eigen)n(v)-5 b(alue,)25 b(pro)n(vided)g(the)h(asso)r (ciated)456 1784 y(eigen)n(v)-5 b(alue)24 b(problem)g(has)g(a)h(p)r (ositiv)n(e)f(solution.)36 b(F)-7 b(urther,)25 b(in)g(the)h(next)f (subsection,)g(w)n(e)f(will)456 1884 y(consider)i(Mark)n(o)n(v)f (shifts)k(for)e(whic)n(h)g(this)h(eigen)n(v)-5 b(alue)27 b(problem)g(is)g(a)g(linear)g(one.)456 2041 y(Assume)34 b(that)h Fq(N)1033 2011 y Fo([)p Fs(k)q Fo(])1146 2041 y Fv(:)g Ft(Z)1266 1998 y Fo(\(2)p Fs(;)o Fo(1\))1266 2051 y Fs(p)1437 2041 y Fr(!)f(f)p Fv(0)p Fq(;)14 b Fv(1)p Fr(g)p Fv(,)35 b(with)g Fq(k)j Fr(2)d(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(g)p Fv(,)36 b(are)d(comm)n(uting)i(irre-)456 2157 y(ducible)f(matrices)f(suc)n(h)h (that,)i(considered)d(as)h(c)n(haracteristic)e(functions)i(on)g Ft(Z)3079 2114 y Fo(\(2)p Fs(;)o Fo(1\))3079 2167 y Fs(p)3215 2157 y Fv(,)i(their)456 2273 y(asso)r(ciated)26 b(sets)h(form)f(a)h (partition.)37 b(Hence)27 b(w)n(e)g(ha)n(v)n(e)f Ft(Z)2276 2230 y Fo(\(2)p Fs(;)p Fo(1\))2276 2283 y Fs(p)2435 2273 y Fv(=)2523 2211 y Fm(S)2552 2273 y Fr(\001)2603 2231 y Fs(K)2603 2298 y(k)q Fo(=1)2728 2273 y Fq(C)2787 2285 y Fs(k)2828 2273 y Fv(,)h(with)h Fq(C)3126 2285 y Fs(k)3190 2273 y Fr(\021)23 b(f)p Fy(a)g Fr(2)456 2393 y Ft(Z)517 2350 y Fo(\(2)p Fs(;)p Fo(1\))517 2403 y Fs(p)676 2393 y Fv(:)51 b Fq(N)826 2363 y Fo(\()p Fs(k)q Fo(\))918 2393 y Fv(\()p Fy(a)p Fv(\))24 b(=)f(1)p Fr(g)p Fv(.)456 2538 y(T)-7 b(o)21 b(eac)n(h)g Fq(k)26 b Fr(2)d(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(g)21 b Fv(w)n(e)g(asso)r(ciate)f(a)i(nonempt)n(y)f(set)h Fq(S)2465 2550 y Fs(k)2529 2538 y Fr(\032)g(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)f(;)h(K)6 b Fr(g)20 b Fv(satisfying)456 2638 y Fq(S)507 2650 y Fs(k)566 2638 y Fr(\\)f Fq(S)691 2650 y Fs(`)746 2638 y Fr(6)p Fv(=)j Fr(;)27 b Fv(for)g(eac)n(h)g Fq(k)s(;)14 b(`)23 b Fr(2)g(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(g)p Fv(.)456 2783 y(No)n(w,)27 b(let)h(us)f(de\014ne)h(the)g(set)g(of)f(admissible)g(arra)n(ys)e(as) 505 2974 y Fr(P)k(\021)680 2857 y Fm(\032\024)839 2923 y Fy(a)827 3023 y(a)873 2993 y Fp(0)939 2857 y Fm(\025)1006 2974 y Fr(2)23 b Ft(Z)1145 2940 y Fo(\(2)p Fs(;)p Fo(2\))1145 2995 y Fs(p)1304 2974 y Fv(:)51 b Fq(N)1454 2940 y Fo(\()p Fs(k)q Fo(\))1547 2974 y Fv(\()p Fy(a)p Fv(\))p Fq(N)1733 2940 y Fo(\()p Fs(`)p Fo(\))1817 2974 y Fv(\()p Fy(a)1895 2940 y Fp(0)1919 2974 y Fv(\))24 b(=)e(1)55 b(for)27 b(some)g(1)c Fr(\024)f Fq(k)k Fr(\024)d Fq(K)33 b Fv(and)27 b Fq(`)c Fr(2)g Fq(S)3255 2986 y Fs(k)3296 2857 y Fm(\033)3372 2974 y Fq(:)456 3227 y Fv(W)-7 b(e)19 b(assume)f(that)h(for)f(eac)n(h)g Fq(k)26 b Fr(2)d(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(g)p Fv(,)19 b(among)f(all)g(the)i(matrices)d(in)i Fr(f)p Fq(N)3018 3196 y Fo([)p Fs(`)p Fo(])3110 3227 y Fv(:)42 b Fq(`)23 b Fr(2)g Fq(S)3362 3239 y Fs(k)3403 3227 y Fr(g)456 3326 y Fv(there)f(are)h(t)n(w)n(o)f(whose)g(p)r(erio)r (ds)h(are)f(relativ)n(e)g(primes.)35 b(This)23 b(ensures)f(that)i(the)f (Mark)n(o)n(v)e(shift)456 3426 y Fq(X)525 3438 y Fp(P)608 3426 y Fv(is)28 b(horizon)n(tally)d(transitiv)n(e.)36 b(It)28 b(is)g(also)e(easily)h(v)n(eri\014ed)g(that)h Fq(X)2659 3438 y Fp(P)2742 3426 y Fv(is)g(fully)g(extensible.)456 3571 y(The)35 b(resulting)g(Mark)n(o)n(v)f(shift)i(ma)n(y)f(b)r(e)h (considered)f(as)g(a)g(conditioned)g(pro)r(duct)h(of)f(one{)456 3671 y(dimensional)29 b(Mark)n(o)n(v)e(shifts.)43 b(Eac)n(h)28 b(matrix)h Fq(N)2033 3641 y Fo([)p Fs(k)q Fo(])2111 3671 y Fq(;)43 b(k)29 b Fr(2)e(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fr(g)28 b Fv(de\014nes)i(a)f(Mark)n(o)n(v) 456 3774 y(shift)c Fq(X)716 3744 y Fo([)p Fs(k)q Fo(])794 3774 y Fv(.)36 b(Lo)r(cally)-7 b(,)25 b(the)h(horizon)n(tal)d (transitions)h Fq(N)2172 3744 y Fo([)p Fs(k)q Fo(])2276 3774 y Fv(and)h Fq(N)2511 3744 y Fo([)p Fs(`)p Fo(])2605 3774 y Fv(ma)n(y)g(co)r(exist)f(if)i(and)f(only)456 3873 y(if)j Fq(`)22 b Fr(2)i Fq(S)719 3885 y Fs(k)760 3873 y Fv(.)36 b(Hence,)28 b(lo)r(cally)f(it)h(corresp)r(onds)e(to)h(a)h (pro)r(duct)f Fq(X)2449 3843 y Fo([)p Fs(k)q Fo(])2545 3873 y Fr(\002)18 b Fq(X)2704 3843 y Fo([)p Fs(`)p Fo(])2773 3873 y Fv(,)28 b(with)g Fq(`)23 b Fr(2)g Fq(S)3200 3885 y Fs(k)3241 3873 y Fv(.)456 4030 y(F)-7 b(or)40 b Fq(X)687 4042 y Fp(P)784 4030 y Fv(de\014ned)i(as)e(b)r(efore)h(w)n(e)g(ha)n(v)n (e)f(represen)n(tativ)n(e)f(matrices)i Fq(M)2787 3987 y Fo([)p Fs(k)q Fo(])2778 4053 y(1)2911 4030 y Fr(\021)3021 3968 y Fm(P)3108 4055 y Fs(`)p Fp(2)p Fs(S)3222 4064 y Fi(k)3276 4030 y Fq(N)3352 4000 y Fo([)p Fs(`)p Fo(])3421 4030 y Fv(,)456 4146 y(with)g Fq(k)47 b Fv(=)e(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i(K)6 b Fv(.)75 b(These)41 b(matrices)e(inherit)i(from)f Fr(f)p Fq(N)2544 4116 y Fo([)p Fs(k)q Fo(])2667 4146 y Fv(:)85 b(1)44 b Fr(\024)h Fq(k)i Fr(\024)d Fq(K)6 b Fr(g)40 b Fv(the)456 4249 y(comm)n(utativit)n (y)-7 b(.)33 b(Their)20 b(irreducibilit)n(y)f(follo)n(ws)g(form)h(the)g (irreducibilit)n(y)f(of)h Fq(N)2979 4219 y Fo([)p Fs(k)q Fo(])3077 4249 y Fv(for)f(eac)n(h)g Fq(k)s Fv(.)456 4365 y(F)-7 b(urthermore,)20 b(since)f Fq(S)1200 4377 y Fs(k)1244 4365 y Fr(\\)s Fq(S)1353 4377 y Fs(k)1389 4360 y Fj(0)1436 4365 y Fv(for)g(all)g Fq(k)s(;)14 b(k)1791 4334 y Fp(0)1837 4365 y Fr(2)24 b(f)p Fv(1)p Fq(;)14 b(:)g(:)g(:)e(;)i(K)6 b Fr(g)p Fv(,)21 b(it)f(follo)n(ws)f(that)h Fq(M)2948 4321 y Fo([)p Fs(k)q Fo(])2939 4387 y(1)3028 4365 y Fr(\003)s Fq(M)3163 4321 y Fo([)p Fs(k)3218 4296 y Fj(0)3240 4321 y Fo(])3154 4387 y(1)3286 4365 y Fr(6)p Fv(=)j Fy(0)o Fv(.)456 4515 y(In)31 b(this)g(example)f(the)h(linking)g(matrices)f (are)g Fq(L)2015 4484 y Fo([)p Fs(k)q(;`)p Fo(])2169 4515 y Fv(=)44 b(I)-50 b(1)2300 4527 y Fs(S)2341 4536 y Fi(k)2381 4515 y Fv(\()p Fq(`)p Fv(\))21 b Fr(\002)f Fq(N)2662 4484 y Fo([)p Fs(`)p Fo(])2731 4515 y Fv(.)47 b(It)31 b(follo)n(ws)f(directly)456 4614 y(that)d(these)h(matrices)f (comm)n(ute.)456 4805 y(Since)41 b(all)f(the)i(desired)e(h)n(yp)r (otheses)g(hold,)45 b(w)n(e)40 b(can)h(compute)g(the)g(top)r(ological)f (en)n(trop)n(y)456 4905 y Fq(h)p Fv(\()p Fq(X)605 4917 y Fp(P)661 4905 y Fv(\))28 b(of)f(this)h(Mark)n(o)n(v)d(shift,)k(pro)n (vided)1062 5132 y Fq(\025)18 b Fr(\002)g Fq(M)9 b Fv(\()p Fy(a)p Fv(\))24 b(=)1523 4990 y Fm(")1571 5132 y Fq(\032)1628 4990 y Fm( )1724 5028 y Fs(K)1694 5053 y Fm(X)1694 5232 y Fs(k)q Fo(=1)1828 5132 y Fq(N)1904 5098 y Fo([)p Fs(k)q Fo(])1982 5132 y Fv(\()p Fy(a)p Fv(\))2106 4990 y Fm( )2188 5053 y(X)2173 5232 y Fs(`)p Fp(2)p Fs(S)2287 5241 y Fi(k)2336 5132 y Fq(M)j Fr(\003)18 b Fq(N)2580 5098 y Fo([)p Fs(k)q Fo(])2659 4990 y Fm(!)o(!#)p eop %%Page: 9 9 9 8 bop 752 251 a Fo(EIGENV)-7 b(ALUE)29 b(EQUA)-5 b(TION)28 b(F)n(OR)i(THE)f(ENTR)n(OPY)g(OF)g(2{D)g(MARK)n(O)n(V)g(SHIFTS)263 b(9)456 450 y Fv(has)33 b(a)g(solution)g(in)h(\001)1177 420 y Fo(\()p Fs(p;p)p Fo(\))1321 450 y Fv(.)55 b(F)-7 b(or)33 b(matrices)g(of)g(the)h(kind)g Fq(M)42 b Fv(=)2555 388 y Fm(P)2643 408 y Fs(K)2643 475 y(k)q Fo(=1)2782 450 y Fq(x)2829 462 y Fs(k)2870 450 y Fq(N)2946 420 y Fo([)p Fs(k)q Fo(])3024 450 y Fv(,)36 b(this)d(prob-)456 550 y(lem)d(reduces)e(to)i(the)g Fq(K)6 b Fv({dimensional)28 b(linear)h(eigen)n(v)-5 b(alue)29 b(problem)g(for)g(the)h(non{negativ)n (e)456 649 y(matrix)560 819 y Fq(S)611 831 y Fp(P)690 819 y Fv(:)23 b Fr(f)p Fv(1)p Fq(;)14 b Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i (K)6 b Fr(g)18 b(\002)g(f)p Fv(1)p Fq(;)c Fv(2)p Fq(;)g(:)g(:)g(:)e(;)i (K)6 b Fr(g)22 b(!)h Ft(R)p Fq(;)47 b Fv(suc)n(h)28 b(that)g Fq(S)5 b Fv(\()p Fq(k)s(;)14 b(`)p Fv(\))22 b(=)39 b(I)-50 b(1)2773 831 y Fs(S)2814 840 y Fi(k)2854 819 y Fv(\()p Fq(`)p Fv(\))19 b Fr(\002)f Fq(\032)p Fv(\()p Fq(N)3206 785 y Fo([)p Fs(k)q Fo(])3284 819 y Fv(\))p Fq(:)456 989 y Fv(If)27 b(w)n(e)f(further)h(assume)f(that)h(this)h(matrix)e(is)h (irreducible,)f(b)n(y)h(the)g(P)n(erron{F)-7 b(rob)r(enius)24 b(The-)456 1089 y(orem)i(w)n(e)i(ha)n(v)n(e)e(a)h(p)r(ositiv)n(e)g (solution.)37 b(In)28 b(this)f(case)456 1259 y(\(19\))993 b Fq(h)p Fv(\()p Fq(X)1746 1271 y Fp(P)1802 1259 y Fv(\))23 b(=)g(log)14 b Fq(\032)p Fv(\()p Fq(S)2192 1271 y Fp(P)2248 1259 y Fv(\))p Fq(:)456 1479 y Fv(3.5.)40 b Fy(Examples.)f Fv(Let)19 b Fr(f)p Fq(P)1329 1491 y Fo(1)1366 1479 y Fq(;)14 b(P)1456 1491 y Fo(2)1494 1479 y Fq(;)g(:)g(:)g(:)f(;)h(P)1731 1491 y Fs(M)1805 1479 y Fr(g)k Fv(b)r(e)h(a)f(collection)g(of)h(comm)n (uting)f(irreducible)g(p)r(er-)456 1579 y(m)n(utations)27 b(in)h Ft(Z)1004 1591 y Fs(p)1036 1579 y Fv(,)g(suc)n(h)f(that)1302 1749 y(\()p Fq(i)22 b Fr(6)p Fv(=)h Fq(j)5 b Fv(\))23 b Fr(\))g Fv(\()p Fq(P)1758 1761 y Fs(i)1787 1749 y Fv(\()p Fq(a)p Fv(\))g Fr(6)p Fv(=)g Fq(P)2059 1761 y Fs(j)2094 1749 y Fv(\()p Fq(a)p 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Fm(p)p 2534 3999 443 4 v 73 x Fv(4)p Fq(M)9 b(p)17 b Fr(\000)h Fv(3)p Fq(M)2940 4048 y Fo(2)p 2260 4109 717 4 v 2597 4185 a Fv(2)2986 3986 y Fm(!)3066 4128 y Fq(:)1554 4468 y Fv(4.)41 b Fu(Final)31 b(Comments)456 4717 y Fv(The)36 b(class)f(of)i(examples)e(w)n(e)h(presen)n(ted)g(are)f (quite)i(restrictiv)n(e)e(in)h(nature.)63 b(It)36 b(w)n(ould)g(b)r(e) 456 4817 y(in)n(teresting)24 b(to)g(construct)h(examples)f(of)h(other)f (kind)h(satisfying)f(all)g(the)i(conditions)e(of)h(The-)456 4917 y(orem)32 b(1.)54 b(In)33 b(particular,)h(it)g(w)n(ould)f(b)r(e)g (in)n(teresting)g(to)g(pro)n(vide)f(examples)h(for)g(whic)n(h)g(the)456 5016 y(en)n(trop)n(y)23 b(is)i(a)g(nonlinear)f(eigen)n(v)-5 b(alue.)35 b(It)25 b(is)g(w)n(orth)f(to)h(men)n(tion)g(that)g(the)g (class)f(of)h(examples)456 5116 y(w)n(e)h(sho)n(w)n(ed)g(in)i(the)f (previous)f(section)h(are)f(related,)h(though)f(not)i(of)f(the)g(same)g (kind,)g(to)g(the)456 5216 y(Matrix)g(subshifts)h(in)n(tro)r(duced)f (in)h([5)o(].)p eop %%Page: 10 10 10 9 bop 456 251 a Fo(10)1102 b(EDGARDO)23 b(UGALDE)1708 450 y Fu(References)491 583 y Fx([1])35 b(A.)18 b(Berman)h(and)g(R.)g (J.)g(Plemmons.)e Fd(Nonne)l(gative)k(Matric)l(es)g(in)h(the)f (Mathematic)l(al)i(Scienc)l(e)p Fx(,)c(Classics)601 666 y(in)k(Applied)h(Mathematics)f Fc(9)p Fx(,)g(SIAM)h(\(1994\).)491 749 y([2])35 b(R.)16 b(Bo)n(w)n(en.)i Fd(Equilibrium)i(States)g(and)g (Er)l(go)l(dic)h(The)l(ory)f(of)g(A)n(nosov)h(Di\013e)l(omorphisms)p Fx(,)f(Lecture)f(Notes)601 832 y(in)k(Mathematics)h Fc(470)p Fx(,)e(Springer)h(\(1975\).)491 915 y([3])35 b(J.)19 b(Juang,)i(S.{S.)e(Lin)g(and)i(S.)e(F.)g(Shieh.)g(The)h(spatial)g(en)n (trop)n(y)h(of)e(t)n(w)n(o{dimensional)g(subshifts)g(of)h(\014nite)601 998 y(t)n(yp)r(e.)k Fd(International)j(Journal)g(of)f(Bifur)l(c)l (ation)h(and)f(Chaos)f Fc(10)e Fx(\(12\))i(\(2000\),)g(2845{2852.)491 1081 y([4])35 b(D.)26 b(Lind,)i(K.)f(Sc)n(hmidt)g(and)h(T.)f(W)-6 b(ard.)27 b(Malher)g(measures)g(and)h(en)n(trop)n(y)g(for)f(comm)n (uting)f(automor-)601 1164 y(phisms)c(of)h(compact)h(groups.)g Fd(Inventiones)h(Mathematic)l(ae)g Fc(101)d Fx(\(1990\),)k(593{629.)491 1247 y([5])35 b(N.)26 b(G.)h(Markley)g(and)g(M.)g(E.)f(P)n(aul.)h (Matrix)f(subshifts)h(for)f Fb(Z)2312 1224 y Fa(\027)2371 1247 y Fx(sym)n(b)r(olic)g(dynamics.)g Fd(Pr)l(o)l(c)l(e)l(e)l(dings)31 b(of)601 1330 y(the)25 b(L)l(ondon)i(Mathematic)l(al)g(So)l(ciety)c Fc(43)g Fx(\(3\))i(\(1981\),)g(251{272.)491 1413 y([6])35 b(R.)18 b(Meester)g(and)h(J.)g(E.)e(Steif.)i(Higer)e(dimensional)h (subshift)g(of)g(\014nite)h(tip)r(e,)h(factor)f(maps)e(and)i(measures) 601 1496 y(of)k(maximal)e(en)n(trop)n(y)-6 b(,)25 b Fd(Paci\014c)h (Journal)h(of)f(Mathematics)e Fc(200)e Fx(\(2\))j(\(2001\),)g(497{510.) 491 1579 y([7])35 b(K.)22 b(Sc)n(hmidt.)h Fd(A)n(lgebr)l(aic)i(Ide)l (as)i(in)e(Er)l(go)l(dic)h(The)l(ory)p Fx(,)e(CBMS)f(Regional)g (Conference)h(Series)f(in)g(Math-)601 1662 y(ematics)g Fc(76)p Fx(,)f(American)h(Mathematical)h(So)r(ciet)n(y)h(\(1990\).)555 1818 y Fw(I)r(ICO{UASLP,)h(A)-8 b(v.)25 b(Karak)o(or)o(um)e(1470,)i (78210)e(San)h(Luis)i(Potos)2498 1812 y(\023)2506 1818 y(\020,)h(M)2640 1812 y(\023)2640 1818 y(exico)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0209111128157--