Content-Type: multipart/mixed; boundary="-------------0201161541433" This is a multi-part message in MIME format. ---------------0201161541433 Content-Type: text/plain; name="02-22.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-22.keywords" tilings, mixing, substitution, subshifts, conjugacy ---------------0201161541433 Content-Type: application/x-tex; name="Final.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Final.tex" %% rec'd 1/4/02 -- changed from {article} to {amsart} style \documentclass[12pt]{amsart} \usepackage{amsmath,amssymb,amscd,amsfonts} %\usepackage{epsf} %\usepackage{FIG} \pagenumbering{arabic} %%%%%%% PAGE STYLE/SIZING %%%%%%%%%%%% \textwidth=6.25truein \textheight=8.5truein \hoffset=-.5truein \voffset=-.5truein \headheight=7pt %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{prop}{Proposition}[section] \newtheorem{thm}[prop]{Theorem} \newtheorem{cor}[prop]{Corollary} \newtheorem{con}[prop]{Conjecture} \newtheorem{Question}[prop]{Question} \newtheorem{rem}[prop]{Remark} \newtheorem{lem}[prop]{Lemma} \def\demo#1{\medskip\noindent{\bf#1\enspace}} \def\enddemo{\medskip} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\setlength{\topmargin}{0in} %\setlength{\textheight}{9in} %\setlength{\oddsidemargin}{0in} %\setlength{\textwidth}{6in} %\pagestyle{empty} \def \ted{{\mathbb T}} \def \ded{{\mathbb D}} \def \l{\ell} \def \led{\mathcal{L}} \def \calS {{\mathcal S}} \def \calT {{\mathcal T}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\B{{\cal B}} %% ?? was this what you wanted? \def\ced{{\cal C}} \def\zed{{\mathbb Z}} \def\red{{\mathbb R}} \def\ied{{\mathbb I}} \def\qed{{\mathbb Q}} \def\alf{\alpha} \def\goesto{\mapsto} %% Hope this is what you want. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{When size matters: subshifts and their related tiling spaces} \author{Alex Clark and Lorenzo Sadun} \address{Alex Clark: Department of Mathematics, University of North Texas, Denton, Texas 76203} \email{AlexC@unt.edu} \address{Lorenzo Sadun: Department of Mathematics, The University of Texas at Austin, Austin, TX 78712-1082 U.S.A.} \email{sadun@math.utexas.edu} \subjclass{52C23, 37A25, 37A30, 37A10, 37B10} %\dedicatory{} \begin{abstract} We investigate the dynamics of substitution subshifts and their associated tiling spaces. For a given subshift, the associated tiling spaces are all homeomorphic, but their dynamical properties may differ. We give criteria for such a tiling space to be weakly mixing, and for the dynamics of two such spaces to be topologically conjugate. \end{abstract} \maketitle \markboth{Clark-Sadun}{When size matters: subshifts and tiling spaces} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} We consider the dynamics of 1-dimensional minimal substitutions subshifts (with a natural $\mathbb{Z}$ action) and the associated 1-dimensional tiling spaces (with natural $\mathbb{R}$ actions). Given an alphabet $\mathcal{A}$ of $n$ symbols $\left\{ a_{1},...,a_{n}\right\} $, a \emph{substitution }on $ \mathcal{A}$ is a function $\sigma $ from $\mathcal{A}$ into the non-empty, finite words of $\mathcal{A}$. Associated with such a substitution is the $ n\times n$ matrix $M$ which has as its $\left( i,j\right) $ entry the number of occurrences of $a_{j}$ in $\sigma \left( a_{i}\right) $. A substitution is \emph{primitive }if some positive power of $M$ has strictly positive entries. The substitution $\sigma $ induces the map $\sigma :\mathcal{A}^{ \mathbb{Z}}\rightarrow \mathcal{A}^{\mathbb{Z}}$ given by $\left\langle \dots u_{-1}.u_{0}u_{1}\dots \right\rangle \overset{\sigma }{\mapsto } \left\langle \dots \sigma \left( u_{-1}\right) .\sigma \left( u_{0}\right) \sigma \left( u_{1}\right) \dots \right\rangle .$ For any primitive substitution $\sigma $ there is at least one point $u\in \mathcal{A}^{ \mathbb{Z}}$ which is periodic under $\sigma ,$ and the closure of the orbit of any such $u$ under the left shift map $s$ of $\mathcal{A}^{\mathbb{Z}}$ forms a minimal subshift $\mathcal{S}$. This subshift $\mathcal{S}$ is uniquely determined by $\sigma ,$ and the restriction $\sigma |_{\mathcal{S}} $ is a homeomorphism onto $\mathcal{S}.$ To avoid trivialities, we shall only consider \emph{aperiodic} substitutions $\sigma ,$ those for which the subshift $\mathcal{S}$ is not periodic. Given a collection of intervals $\mathcal{I}=\left\{ I_{1},...,I_{n}\right\} $, a \emph{tiling} $T$ of $\mathbb{R}$ by $\mathcal{I}$ is a collection of closed intervals $\left\{ T_{i}\right\} _{i\in \mathbb{Z}}$ satisfying \begin{enumerate} \item $\cup _{i\in \mathbb{Z}}T_{i}=\mathbb{R},$ \item For each $i\in \mathbb{Z},$ $T_{i}$ is the translate of some $I_{\tau \left( i\right) }\in \mathcal{I},$ and \item $T_{i}\cap T_{i+1}$ is a singleton for each $i\in \mathbb{Z}.$ \end{enumerate} \noindent If the function $\tau :\mathbb{Z\rightarrow }\left\{ 1,...,n\right\} $ is an element of a minimal substitution subshift $\mathcal{ S}$ of $\left\{ 1,...,n\right\} ^{\mathbb{Z}}$, then $T$ is called a \emph{substitution tiling.} There is a natural topology on the space $\mathfrak{T}$ of tilings of $\mathbb{R}$ that is induced by a metric which measures as close any two tilings $T$ and $T^{\prime }$ that agree on a large neighborhood of $0$ up to an $\varepsilon $ translation. There is then the continuous translation action $\mathbf{T}$ of $\mathbb{R}$ on $\mathfrak{T:}$ for $t\in \mathbb{R}$ and the tiling $T=\left\{ T_{i}\right\} _{i\in \mathbb{ Z}}$, $\mathbf{T}:\left( T,t\right) \mapsto \mathbf{T}_{t}T=\left\{ T_{i}-t\right\} _{i\in \mathbb{Z}}.$ (A positive element of $\red$ moves the origin to the right, or equivalently moves tiles to the left). The closure of the translation orbit of any substitution tiling $T$ in $\mathfrak{T}$ is then a minimal set of the action, the (\emph{substitution})\emph{\ tiling space }$\mathcal{T}$ of the tiling $T$. As the tiling space for any iterate of a substitution $\sigma $ is the same as the tiling space of $\sigma $ and since any substitution has a point $u$ whose right half $\left\langle u_{0}u_{1}\dots \right\rangle $ is periodic under substitution, for ease of discussion we shall only consider substitutions with a point $u$ whose right half $\left\langle u_{0}u_{1}\dots \right\rangle $ is fixed under substitution. Flows under a function provide an alternative description of tiling spaces convenient for our purposes. Given a minimal subshift $\left( \mathcal{S} ,s\right) $ of the shift on $\mathcal{A}^{\mathbb{Z}}$ and $f:\mathcal{A} \rightarrow \left( 0,\infty \right) ,$ there is the flow under $f$ given by the natural $\mathbb{R}$ action on $\calT_{f}=\calS\times \mathbb{R}/\sim $, where $(u,f(u_{0}))\sim (s(u),0)$. If we then associate with each $a_{i}\in \mathcal{A}$ a closed interval $I_{i}$ of length $f\left( a_{i}\right) $ and form the tiling on $T$ of $\mathbb{R}$ by $\left\{ I_{1},\dots ,I_{n}\right\} $ with associated function $\tau =u\in \mathcal{A}^{\mathbb{Z} }$, which has the left endpoint of the interval corresponding to $u_{0}$ at $ 0\in \mathbb{R}$, then the function sending $T$ to the class of $\left( u,0\right) $ in $\calT_{f}$ extends uniquely to a homeomorphism $\mathcal{T} \rightarrow \mathcal{T}_{f}$ which conjugates the respective $\mathbb{R}$ actions. The primary focus of this paper is the extent to which the dynamical systems $\mathcal{T}_{f}$ depend on the function $f$. If all we care about is the topological space, they don't: \begin{thm} \label{homeo}Let $\calS$ be a subshift, and let $f,g$ be two positive functions on the alphabet of $\calS$. Then $\calT_{f}$ and $\calT_{g}$ are homeomorphic. \end{thm} \medskip \noindent \textbf{Proof.\enspace} Every class $x\in \calT_{f}$ has a (unique) representative of the form $(u,t)$, with $0\leq t0$ is the same, only with $\calT_f$ and $\calT _g$ reversed. If (\ref{samesize}) holds with $k=0$, then the supertiles in the $\calT_{f}$ system asymptotically have the same size as the corresponding supertiles in the $\calT_{g}$ system, and the convergence is exponential. We then adapt the argument that Radin and Sadun \cite{RS} applied to the Fibonacci tiling. If $x$ is a tiling in $\calT_{f}$ with tiles in a sequence $u=\ldots u_{-1}u_{0}u_{1},\ldots $, then we require $\phi _{\ell }(x)\in \calT_{g}$ to be a tiling with the exact same sequence of tiles. The only question is where to place the origin. Let $d_{\ell }$ be the coordinate of the right edge of the order-$\ell $ supertile of $x$ that contains the origin, and let $-e_{\ell }$ be the coordinate of the left edge. Place the origin in $\phi _{\ell }(x)$ a fraction $e_{\ell }/(d_{\ell }+e_{\ell })$ of the way across the corresponding supertile of order $\ell $ in the $\calT_{g}$ system. (This is essentially the map $h$ of Theorem \ref{homeo}, only applied to supertiles of order $\ell $.) Since the sizes of the supertiles converge exponentially, the location of the origin in $\phi _{\ell }(x)$ converges, and we can define $\phi (x)=\lim_{\ell \rightarrow \infty }\phi _{\ell }(x)$ . All that remains is to show that $\phi $ is a conjugacy. The approximate map $\phi _{\ell }$ is not a conjugacy; translations in $x$ the keep the origin in the same supertile of order $\ell $ get magnified (in $\phi (x)$) by the ratio of the sizes of the supertiles of order $\ell $ containing the origin in $x$ and $\phi (x)$. However, this ratio goes to one as $\ell \rightarrow \infty $, while the range of translations to which this ratio applies grows exponentially. In the $\ell \rightarrow \infty $ limit, $\phi $ commutes with translation by any $s\in (-e_{\infty },d_{\infty })$. Typically this is everything. The case where $x$ contains two infinite-order supertiles is only slightly trickier -- once one sees that $\phi $ preserves the boundary between these infinite-order supertiles, it is clear that $\phi $ commutes with all translations. Now suppose that $L=L^{\prime }M$. Then the tiles in the $\calT_{f}$ system have exactly the same size as the corresponding supertiles of order 1 in the $\calT_{g}$ system. We define our conjugacy $\phi $ as follows: If $x$ is a tiling in $\calT_{f}$ with tiles in a sequence $u=\ldots u_{-1}u_{0}u_{1}\ldots $, then $\phi (x)$ to a tiling in $\calT_{g}$ with sequence $\sigma (u)$, with each tile in $x$ aligned with the corresponding order-1 supertile in $\phi (x)$. Combining the cases, we obtain conjugacy whenever (\ref{samesize}) applies.\hfill $\square $\medskip Theorem \ref{sufficient} gives sufficient conditions for two tiling spaces to be conjugate. To derive necessary conditions we must understand the extent to which words in our subshift $\calS$ (and its associated tiling spaces) can repeat themselves. Moss\'{e} shows in \cite{M1} that in every substitution subshift there exists an integer $N$ so that no word is ever repeated $N$ or more times. We will need to refine these results by considering words that are repeated a fractional number of times. Counting fractional periods depends on the length vector: the word $ababa$ has its basic period $ab$ repeated $2+f(a)/(f(a)+f(b))$ times. \noindent \textbf{Definition.} A recurrence vector $v$ is a \textit{ repetition vector of degree }$p$\textit{\ }of a tiling space $\calT$ if there are finite words $w$ (with population vector $v$) and $w^{\prime }$, such that: 1) $w^{\prime }$ contains $w$, 2) $w^{\prime }$ is periodic with period equal to the length of $w$, 3) the length of $w^{\prime }$ is at least $p$ times the length of $w$, and 4) $w^{\prime }$ appears in some (and therefore every) tiling in $\calT$.\smallskip Note that every repetition vector of degree $p>p^{\prime}$ is also a repetition vector of degree $p^{\prime}$. Note also that the word $w$ is typically not uniquely defined by $v$. A cyclic permutation of the tiles in $ w$ typically yields a word that works as well. For instance, if $ w^{\prime}=ababa$ appears in a tiling, then $v=(1,1)^T$ is a repetition vector with period $2+ f(a)/(f(a)+f(b))$, and we may take either $w=ab$ or $ w=ba$. We denote the length $L_{0}w$ of a word $w$ in $\calT_{0}$ as described above by $|w|$. If $v$ is a repetition vector of degree $p$ for $\calT_{0}$, representing the word $w$ sitting inside $ w^{\prime }$ in $u\in \mathcal{A}^{\mathbb{Z}}$, then $Mv$ is a repetition vector of degree $p$, representing the word $\sigma (w)$ sitting inside $ \sigma (w^{\prime })$ in $\sigma (u)\in \mathcal{A}^{\mathbb{Z}}$. The degree is the same, since in $\calT_{0}$ the substitution $\sigma $ stretches each word by exactly the same factor, namely $\lambda _{PF}$. Thus, every repetition vector $v$ gives rise to an infinite family of repetition vectors $M^{k}v$. The following theorem limits the number of such families. \begin{thm} Let $p>1$. There is a finite collection of vectors $\left\{ v_{1},\ldots ,v_{N}\right\} $ such that every repetition vector of degree $p$ for $\calT _{0}$ is of the form $M^{k}v_{i}$ for some pair $(k,i)$. \label{families} \end{thm} \medskip \noindent \textbf{Proof.\enspace} There is a recognition length $D_{ \mathcal{S}}$ of the subshift $\calS$ such that knowing a letter, its $D_{_{ \mathcal{S}}}$ immediate predecessors and its $D_{_{\mathcal{S}}}$ immediate successors determines the supertile of order 1 containing that letter, and the position within that supertile of the letter. Thus for every substitution tiling space $\calT$ there is a recognition length $D_{\mathcal{ T}}$ such that the neighborhood of radius $D_{_{\mathcal{T}}}$ about a point determines the supertile of order 1 containing that point, and the position of that point within the supertile. (E.g., one can take $D_{_{\mathcal{T}}}$ to be $1+D_{_{\mathcal{S}}}$ times the size of the largest tile.) Let $D=D_{ \mathcal{T}_{0}}$. Suppose $v$ is a repetition vector of degree $p$ for $\mathcal{T}_{0}$, corresponding to a word $w$ that is repeated $p$ times in a word $w^{\prime } $, whose length is much greater than $D$. Then there is an interval of size $|w^{\prime }|-2D$ in which the supertiles of order 1 are periodic with period $|w|$. Thus $v$ is the population vector of a collection of supertiles of order 1, so there exists a population vector $v_{1}$ of a word $w_{1}$, such that $v=Mv_{1}$, and such that $v_{1}$ is a repetition vector of degree $(|w^{\prime }|-2D)/|w|=p-2D/\lambda _{PF}|w_{1}|$. Note that $w$ itself does not have to equal $\sigma (w_{1})$ --- it may happen that $w$ is a cyclic permutation of $\sigma (w_{1})$. Repeating the process, we find words $w_i$ such that $\sigma(w_i)$ equals $ w_{i-1}$ (up to cyclic permutation of tiles), and such that the population vector of $w_i$ is a repetition vector of degree \begin{equation} p - \frac{2 D}{|w_i|} (\lambda_{PF}^{-1} + \lambda_{PF}^{-2} + \cdots + \lambda_{PF}^{-i}) \ge p - \frac{2 D}{|w_i|} \sum_{\ell=1}^\infty \lambda_{PF}^{-\ell} = p - \frac{2D}{|w_i| (\lambda_{PF}-1)}. \end{equation} Now pick $\epsilon 0$ such that $|r_w - c| < c_1 / |w|^{c_2}$. \end{lem} \medskip \noindent \textbf{\enspace} The proof is trivial.\hfill $\square $ \medskip \begin{lem} Let $\epsilon >0$, and let $L$ and $L^{\prime}$ be fixed. There is a length $ R$ such that, if $v$ is a repetition vector of degree $p$ in $\calT_f$, representing a word $w$ of length greater than $R$, then $v$ is a repetition vector of degree $p-\epsilon$ in $\calT_g$. \label{samerepvecs} \end{lem} \medskip \noindent \textbf{\enspace} Again, trivial.\hfill $\square $\medskip Pick $p_0>1$ such that there exist repetition vectors of degree greater than $p_0$ in $\calT_0$. By Theorem \ref{families}, there are a finite number of vectors $v_i$ such that every repetition vector of degree $p_0$ is of the form $M^k v_i$. Now, for each $k,i$, let $p_{k,i}$ be the maximal degree of $ M^k v_i$, and let $p_i = \lim_{k \to \infty} p_{k,i}$. Since the $p_i$'s are a finite set of real numbers, all greater than $p_0$, there are real numbers $p$ and $\epsilon$ such that $p > p_0 + \epsilon$ and such that each $p_i$ is greater than $p+ \epsilon$. Let $L$, $L^{\prime}$ and $L_0$ be given. By lemma \ref{samerepvecs}, there is a length $R$ such that the set of repetition vectors $v$ of degree $p$ and with $L_0 v>R$ is precisely the same for the three tiling systems $\calT_f$, $\calT_g$ and $\calT_0$. Pick generators $v_1, \ldots, v_N$ of the finite families of these repetition vectors, with $L_0 v_i \le L_0 v_{i+1}$, and with $L_0 v_N < \lambda_{PF} L_0 v_1$. The asymptotic lengths of these repetition vectors are conjugacy invariants: \begin{thm} Suppose that $\calT_{f}$ and $\calT_{g}$ are conjugate and that $\left\{ v_{1},\dots v_{N}\right\} $ is a collection of repetition vectors of degree $ p$ that generates all repetition vectors of degree $p.$ Then, given $\delta >0$, for every $i\in \{1,\ldots ,N\}$ and for any sufficiently large integer $m$, there exist $j,m^{\prime }$ such that $|LM^{m}v_{i}-L^{\prime }M^{m^{\prime }}v_{j}|<\delta $. \label{necessary} \end{thm} \medskip \noindent \textbf{Proof.\enspace} Let $\phi :\calT_{f}\rightarrow \calT_{g}$ be the conjugacy. Since $\calT_{f}$ and $\calT_{g}$ are compact metric spaces, $\phi $ is uniformly continuous. Now suppose $v_{i}$ is a repetition vector of degree $p$. Let $t_{m,i}=LM^{m}v_{i}$. Then for every tiling $x\in \calT_{f}$ there is a range of real numbers $r$ of size roughly $(p-1)t_{m,i}$, such that $T_{r}(x)$ is very close to $T_{r+t_{m,i}}(x)$, where $T_{r}$ denotes translation by $r$. Specifically, as $m\rightarrow \infty $, the distance from $T_{r+t_{m,i}}(x)$ to $T_{r}(x)$ can be made arbitrarily small, and the size of the range of $r$'s, divided by $t_{m,i}$, can be made arbitrarily close to $p-1$. This implies that $\phi (T_{r}(x))$ and $\phi (T_{r+t_{m,i}}(x))$ are extremely close. But $\phi (T_{r}(x))=T_{r}(\phi (x))$ and $\phi (T_{r+t_{m,i}}(x))=T_{r+t_{m,i}}(\phi (x))$, so $\calT_{g}$ admits a repetition vector whose pairing with $ L^{\prime }$ is extremely close to $t_{m,i}$, and whose degree is extremely close to $p$. However, for $m$ large, all such repetition vectors are of the form $M^{m^{\prime }}v_{j}$. \hfill $\square $\medskip As an illustration of the power of this theorem, consider the substitution $ a\rightarrow aaaabb$, $b\rightarrow babbba$. The substitution matrix is $ \left(\begin{smallmatrix} 4 & 2\cr2 & 4 \end{smallmatrix}\right) $, whose eigenvalues, namely 6 and 2, are both greater than one. It is easy to see that $v=(1,0)^{T}$ is a repetition vector of degree 5, and this vector generates the only family of repetition vectors with $p=5$. If $\calT _{f}$ and $\calT_{g}$ are conjugate, for each large $m$ there must exist $ m^{\prime }$ such that $(LM^{m}-L^{\prime }M^{m^{\prime }})v$ is small. Since $v$ is full, this implies there is an integer $k$ such that $L^{\prime }=LM^{k}$. Those are the only possible conjugacies. As a second example, consider the substitution $a\rightarrow aaaabb$, $ b\rightarrow bbbbaa$. This has exactly the same substitution matrix, but the periodicities are different. $v_{1}=(1,0)^{T}$ and $v_{2}=(0,1)^{T}$ are both repetition vectors with degree 6. The possibility of having $j\neq i$ in Theorem \ref{necessary} allows for additional conjugacies. Indeed, it is not hard to see that $\calT_{f}$ and $\calT_{g}$ are conjugate if (and only if) either $L^{\prime }=(L_{1},L_{2})M^{k}$ or $L^{\prime }=(L_{2},L_{1})M^{k}$ for some (possibly negative) integer $k$. The difference between these two examples illustrates that one cannot obtain sharp conditions on conjugacy from the substitution matrix alone. One needs some details about the substitution itself. Another class of examples illustrating the importance of the actual substitution as opposed to its matrix can be constructed using powers of matrices. For instance, the substitution $a\rightarrow aabaabbba,$ $ b\rightarrow bbabbaaab$ has matrix $M= \left(\begin{smallmatrix} 5 & 4\cr4 & 5 \end{smallmatrix}\right) = \left(\begin{smallmatrix} 2 & 1\cr1 & 2 \end{smallmatrix}\right)^{2}$ and is the square of the substitution $a\rightarrow aab,$ $ b\rightarrow bba,$ and so it can be seen that the associated tiling space admits (among others) conjugacies with $L^{\prime }=L \left(\begin{smallmatrix} 2 & 1\cr1 & 2 \end{smallmatrix}\right)^m$, where $m$ can be any integer. But the substitution $a\rightarrow aaaaabbbb,$ $b\rightarrow abbbbbaaa $, with the same matrix $M$, has only one family of repetition vectors of degree $8$, namely those generated by $\left( 1,0\right) ^{T}$. In this case all conjugacies are of the form $L^{\prime }=LM^{k} =L \left(\begin{smallmatrix} 2 & 1 \cr 1 & 2 \end{smallmatrix}\right)^{2k}.$ \bigskip We thank Felipe Voloch, Bob Williams and Charles Radin for helpful discussions. 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