0$ there exists a length $R$ such that all recurrences vectors $v$ with $|\v|_f>R$ have $|r_\v - 1| < \epsilon$. \label{lem3.3} \end{lem} \medskip Recurrences of the form $M^n \v$, with $n$ large, have lengths in $\calT_f$ and $\calT_0$ that differ only slightly. Since any long recurrence is the sum of high order recurrences plus a bounded number of lower order recurrences, the result follows. \QED \medskip \begin{lem} Let $\epsilon >0$, and let $f$ be fixed. There is a self-similar tiling space $\calT_0$ and a length $ R$ such that every recurrence vector of degree at least $p$ in $\calT_f$, of length greater than $R$, is a recurrence vector of degree at least $p-\epsilon$ in $\calT_0$. Furthermore, every recurrence vector of degree at least $p+\epsilon$ in $\calT_0$, of length greater than $R$, is a recurrence vector of degree at least $p$ in $\calT_f$. \label{samerepvecs} \end{lem} \demo{Proof.} Decompose $\L_f=\L_0 + \L_r$ as above, and let $\calT_0$ be the pseudo-self-similar space with length vector $\L_0$. By Lemma \ref{lem3.3}, the ratio of lengths in $\calT_f$ and $\calT_0$ approaches 1 for large recurrences. Thus the ratio of the radii of the balls around $\z_1$ and $\z_2$ and the distance $|\z_2-\z_1|$ are within $\epsilon$ for large recurrences. \QED Theorem \ref{families} showed there there are only a finite number of families of recurrence vectors of a given degree for a self-similar tiling. Lemma \ref{samerepvecs} extends that result to all tiling spaces. We now construct a topological invariant from the asymptotic displacements $\L_f \v$ of large recurrence vectors $\v$. \begin{thm}\label{staysame} Suppose that $\phi: \calT_f \mapsto \calT_g$ is a topological conjugacy. Given any positive constants $p, \epsilon_1, \epsilon_2$, there exists a positive constant $R$ such that, for each recurrence vector $\v$ of degree at least $p$ for $\calT_g$ with $|\v|_f>R$, there exists a recurrence vector $\v'$ of degree at least $p-\epsilon_1$ for $\calT_g$ with $|\L_f \v-\L_g \v'|<\epsilon_2$. \label{invariant} \end{thm} In other words, up to small errors in the degree and the displacement that vanish in the limit of large recurrence classes, the degrees and displacements of the recurrence classes are conjugacy invariants. \demo{Proof.} Let $A$ be the diameter of the largest tile in the $\calT_g$ system. Since $\phi$ is uniformly continuous (being continuous with a compact domain), there exists a constant $D_0$ such that, if $x$ and $y$ are tilings in $\calT_f$ that agree on a ball of radius $D_0$ around the origin, then $\phi(x)$ and $\phi(y)$ agree on a ball or radius $A$ around the origin, up to a small translation. Thus if $x$ and $y$ agree on a ball of radius $R>D_0$, then $\phi(x)$ and $\phi(y)$ agree on a ball of radius $R-D_0$, up to a translation whose norm is bounded by a decreasing function $h(R)$, with $\lim_{R \to \infty} h(R)=0$. Now let $\v$ be a recurrence vector of degree $p$ for $\calT_f$, representing a recurrence $(\z_1,\z_2)$ in a tiling $x \in \calT_f$. Then $x-\z_1$ and $x-\z_2$ agree on a ball of radius $p|\v|_f$ about the origin, so $\phi(x)-\z_1$ and $\phi(x)-\z_2$ agree on a ball of radius $p|\v|_f-D_0$, up to translation of size at most $h(R)$. Thus there exists a point $\z_3$, within $h(D)$ of $\z_2$, such that $\v'=[(\z_1,\z_3)]$ is a recurrence class of degree at least $\frac{(p|\v|_f-D_0}{ |\v|_f+h(D)}$ in $\calT_g$. For $D$ large enough, this is greater than $p-\epsilon_1$ and $|\L_f \v- \L_g \v'| < h(R)$ is less than $\epsilon_2$. \QED \demo{Proof of Theorem \ref{necessary0}.} Note that all recurrence vectors are recurrence vectors of some positive degree. We can therefore pick $p_0$ such that the integer span of the recurrence vectors of degree at least $p_0$ is an $s$-dimensional sublattice of $\Z^s$. Pick $\epsilon$ small enough (and adjust $p_0$ by up to $\epsilon$, if necessary) so that all the families of recurrence vectors of degree at least $p_0-\epsilon$ are also families of recurrence vectors of degree at least $p_0+\epsilon$. Let $\v_1, \ldots, \v_k$ be generating vectors of those families. By multiplying by appropriate powers of $M$, these can be chosen so that all of the magnitudes of the displacements $|M^k \v_i|_f$ are within a factor of $\lambda_{PF}$ of one another for large $k$. Now pick a neighborhood $U_\epsilon$ of $f$ in $C^1(\Gamma, \R^d)$ small enough that, if $g' \in U_\epsilon$, then every recurrence class in $\calT_f$ of degree at least $p_0+\epsilon$ is also a recurrence class in $\calT_{g'}$ of degree at least $p_0$, and such that every recurrence class in $\calT_{g'}$ of degree at least $p_0$ is a recurrence class in $\calT_f$ of degree at least $p_0-\epsilon$. This insures that the families of recurrence classes of degree at least $p_0$ in the two tiling spaces are exactly the same. If $\calT_f$ and $\calT_{g'}$ are conjugate then, by Theorem \ref{invariant}, the displacements $ \L_f M^k \v_j$ can be approximated by $\L_{g'} M^{k'} \v_{j'}$ for some $k', j'$, and the approximation get successively better as $k \to \infty$. However, if $U_\epsilon$ is chosen small enough, the only values of $k',j'$ that come close to approximating are $k'=k$ and $j'=j$. Thus the limit of $(\L_f-\L_{g'}) M^k \v_j$ must be zero. By Theorem 2.3, this implies that $(\sigma^*)^k(\S(f)-\S(g'))$ approaches zero as $k \to \infty$, and hence that $\S(f)-\S(g') \in S(\calT)$. Finally, let $U_f = \S(U_\epsilon)$. If $\S(g) \in U_f$, then $\calT_g$ is MLD to a tiling space $\calT_{g'}$ with $g' \in U_\epsilon$ and $\S(g')=\S(g)$. Since $\calT_{g'}$ is conjugate to $\calT_f$, $\S(f)-\S(g) = \S(f)-\S(g') \in S(\calT)$. \QED %Theorem \ref{staysame} says that, asymptotically, degrees and displacements %of large recurrences are conjugacy invariants. An infinitesimal %deformation must therefore preserve the degrees and lengths of each %family of recurrence vectors. However, a large deformation may achieve %a conjugate tiling space by permuting the recurrence vectors of a given %degree, such that {\em set} of displacements is (asymptotically) preserved, %even though each displacement is changed. (See the one-dimensional %examples in \cite{CS}.) To control this phenomenon, it is sometimes %useful to restrict attention to recurrence vectors of sufficiently high %degree. %Here is our old Theorem \ref{necessary}, which talks about necessary %conditions for length vectors that are not infinitesimally close: I'm %not sure how we want to adjust it to higher dimensions, IF AT ALL. %\begin{thm} Let $p>1$ be chosen as in the previous paragraph. %Suppose that $\calT_{f}$ and $\calT_{g}$ are conjugate and that %$\left\{ \v_{1},\dots \v_{N}\right\} $ is a collection of recurrence %vectors of degree $ p$ that generates all recurrence vectors of degree %$p.$ Then, given $\delta >0$, for every $i\in \{1,\ldots ,N\}$ and for %any sufficiently large integer $m$, there exist $j,m'$ such %that $|LM^{m}\v_{i}-L'M^{m'}\v_{j}|<\delta %$. \label{necessary} %\end{thm} \section{Ergodic Properties} Now we turn to the topological point spectrum of substitution tiling spaces, by which we mean the eigenvalues of continuous eigenfuctions of the translation action. First we determine general criteria for a vector to be an eigenvalue for a continuous eigenfunction, and then we apply this criteria to some special cases, depending on the form of the matrix $M$ defined in Section \ref{sub}. This eventually leads to criteria for topological weak mixing. \begin{thm} The vector $\k\in\R^d$ is in the point spectrum of $\mathcal{T}_{f}$ if and only if, for every recurrence vector $\v$, \begin{equation}\label{eig} \dfrac{1}{2\pi } (\k \cdot \L_f) M^{m}\v\rightarrow 0 \text{(mod 1) as } m\rightarrow \infty , \end{equation} where the convergence is uniform in the size of $\v$. %such that for any $\epsilon > 0$ and for all recurrence vectors $\v$ %of sufficiently large size % %\begin{equation}\label{eps} %\dfrac{1}{2\pi } kLM^{m}\v< \epsilon \text{(mod 1). } %\end{equation} \label{zeromod1}\ \end{thm} \medskip \demo{Proof.} Let $x_0$ be a tiling in $\calT_0$ fixed by the substitution homeomorphism, let $x$ be its image in $\calT_f$ under the homeomorphism of Theorem 1.1, and let $E:\mathcal{T}_{f} \mapsto S^1$ be a continuous eigenfunction with eigenvalue $\k$. Let $\v$ be a recurrence vector, then there is a recurrence $(\z_1,\z_2)$ in $x$ with vector $\v$ of some size $s_0$. By applying the substitution homeomorphism $m$ times, we obtain a recurrence $(\z_1^m,\z_2^m)$ in $x$ with recurrence vector $M^{m}\v$, whose displacement is $\z_2^m - \z_1^m = \L_f M^m \v$. Then $x-\z_1^{m}$ and $x-\z_2^{m}$ agree on patches of size $s_m$, where $s_m \rightarrow \infty$ as $m \rightarrow \infty.$ Hence \begin{equation} 1=\text{lim}_{m \rightarrow \infty} \dfrac{E(x-\z_2^{m})}{E(x-\z_1^{m})} =\text{lim}_{m \rightarrow \infty} \dfrac{E(x)\text{exp}(-i\k \cdot \z_2^{m})} {E(x)\text{exp}(-i\k \cdot \z_1^{m})}= \text{lim}_{m \rightarrow \infty} \text{exp}(-i\k\cdot (\z_2^{m}-\z_1^{m})). \end{equation} Thus, we obtain Equation \ref{eig}, and the uniform convergence follows from the uniform continuity of $E$. Conversely, assume that we have the stated convergence for all recurrence vectors $\v$ for some $\k \in \R^{d}$. We construct a continuous eigenfunction $E$ by first assigning $x$ the value $1$. Then we necessarily have for any $\z \in \R^{d}$, $E(x-\z)=\text{exp}(-i\k \cdot \z)$. To show that $E$ extends as required to all of $\mathcal{T}_{f}$, it suffices to show that $E$ as so defined is uniformly continuous on the orbit of $x$. But, given an $\epsilon >0$, by Equation \ref{eig} if $x-\z_1$ and $x-\z_2$ agree on patches of sufficiently large size up to a small translation, we have that $E(x-\z_1)$ and $E(x-\z_2)$ agree to within $\epsilon$. \QED The application of this criterion depends on the eigenvalues of $M$ and on the possible forms of the recurrence vectors. \begin{thm} Suppose that all the eigenvalues of $M$ are of magnitude 1 or greater. If $\k$ is in the spectrum, then all elements of $\k \cdot \L_f/2\pi$ are rational. \label{allbig1} \end{thm} \medskip \noindent \textbf{Proof.\enspace} Let $\k$ be in the point spectrum, and consider the sequence of real numbers $t_{m}=(\k \cdot \L_f)M^{m}\v/(2\pi )$, where $\v$ is a fixed recurrence vector. Let $p(\lambda )=\lambda ^{n}+a_{n-1}\lambda ^{n-1}+\cdots +a_{0}$ be the characteristic polynomial of $M$. Note that the $a_{i}$'s are all integers, since $M$ is an integer matrix. Since $p(M)=0$, the $t_{m}$'s satisfy the recursion: \begin{equation} t_{m+n}=-\sum_{k=0}^{n-1}a_{k}t_{m+k}. \label{intrecur} \end{equation} By Theorem \ref{zeromod1}, the $t_{m}$'s converge to zero $\pmod{1}$. That is, we can write \begin{equation} t_{m}=i_{m}+r_{m} \label{splits} \end{equation} where the $i_{m}$'s are integers, and the $r_{m}$'s converge to zero as real numbers. By substituting the division (\ref{splits}) into the recursion (\ref{intrecur}), we see that both the $i$'s and the $r$'s must separately satisfy the recursion (\ref{intrecur}), once $m$ is sufficiently large. However, any solution to this recursion relation is a linear combination of powers of the eigenvalues of $M$ (or polynomials in $m$ times eigenvalues to the $m$-th power, if $M$ is not diagonalizable). Since the eigenvalues are all of magnitude one or greater, such a linear combination converges to zero only if it is identically zero. Therefore $r_{m}$ must be identically zero for all sufficiently large values of $m$. Apply this procedure to $s$ linearly independent recurrence vectors $\v_1,\ldots,\v_s$, and pick $m$ large enough that the corresponding $t_m(\v_i)$ are integers for each $i=1,\ldots,s.$ Note that $t_{m}$ is an integer linear combination of the elements of the vector $ \k \cdot \L_f/(2\pi )$. However, \begin{equation} (t_m(\v_1),\ldots,t_m(\v_s)) = (\k \cdot \L_f/2\pi) M^m (\v_1,\ldots, \v_s) \end{equation} The matrices $M$ and $V= (\v_1,\ldots,\v_s)$ are invertible and have integer entries, so by Cramer's rule their inverses have rational entries. Thus the components of $\k \cdot \L_f/(2\pi )$ must all be rational. \hfill $\square$\medskip \begin{cor} Suppose all the eigenvalues of $M$ have magnitude 1 or greater. Let $G \subset \R^d$ be the free Abelian group generated by the entries of $\L_f$. Let $\bar G$ be the closure of $G$ in $\R^d$, and let $P$ be the identity component of $\bar G$. The spectrum of $\calT_f$ lies in the orthogonal complement of $P$. \end{cor} In dimension greater than 1, the assumptions of Theorem \ref{allbig1} are rarely met. When eigenvalues of magnitude less than 1 exist, the conclusions are somewhat weaker. \begin{thm} \label{somesmall1} Let $S$ be the span of the (generalized left-) eigenspaces of $M$ with eigenvalues of magnitude strictly less than 1. If $\k$ is in the point spectrum, then $\k \cdot \L_f/ {2 \pi}$ is the sum of a rational vector and an element of $S$. \end{thm} \medskip\noindent\textbf{Proof.\enspace} Let $V$ be the matrix $(\v_1,\ldots,\v_s)$ as in the proof of Theorem \ref{allbig1}. Construct the row vector \begin{equation} \t_m = \dfrac{1}{2\pi } (\k \cdot \L_f) M^m V= \dfrac{1}{2\pi } (\k \cdot \L_f) V (V^{-1}MV)^m. \end{equation} As in the proof of Theorem \ref{allbig1}, each entry of $\t_m$ converges to zero (mod 1), so we can write $\t_m = \i_m + \r_m$, with each entry of $\i_m$ integral and $\r_m$ converging to zero, and with the eventual conditions \begin{equation} \i_{m+1} = \i_m V^{-1} M V; \hspace{.1in} \r_{m+1} = \r_m V^{-1} M V. \end{equation} Since the $\r_m$'s converge to zero, they must lie in the span of the small eigenvalues of $V^{-1} M V$. Thus, by adding an element of $S$ to $(\k \cdot \L_f)/2\pi$, we can then get all the $\r_m$'s to be identically zero beyond a certain point. Since $M$ is invertible, the resulting value of $\k \cdot \L_f/2\pi$ must then be rational. \QED \medskip Another way of stating the same result is to say that $(\k \cdot \L_f)/2\pi$, projected onto the span of the large eigenvectors, equals the projection of a rational vector onto this span. This theorem can be used in two different ways. First, it constrains the set of shape parameters (that is, vectors $\L_f$) for which the system admits point spectrum. Let $d_b$ be the number of large eigenvalues, counted with (algebraic) multiplicity. There are only a countable number of possible values for the projection of $\k \cdot \L_f/2\pi$ onto the span of the large (generalized) eigenvectors. In other words, one must tune $d_b$ parameters to a countable number of possible values in order to achieve a point in the spectrum. Of course, $\k$ itself gives $d$ parameters. Thus we must tune at least $d_b-d$ additional parameters to have any point spectrum at all. In particular, if $d_b>d$, then a generic choice of shape parameter gives topological weak mixing, proving Theorem \ref{mixing}. (Note that the Perron-Frobenius eigenvector always occurs with multiplicity $d$, so that $d_b-d$ is never negative.) A second usage is to constrain the spectrum for fixed $\L_f$. The rational points in $\red^s$, projected onto the span of the large eigenvalues, and then intersected with the $d$-plane defined by a fixed $\L_f$ (i.e., the set of all possible products $(\k \cdot \L_f)/2\pi$), forms a vector space over $\qed$ of dimension at most $s+d-d_b$. As a result, the point spectrum tensored with $\mathbb{Q}$ is a vector space over $\qed$ whose dimension is bounded by $d$ plus the number of small eigenvalues. Below we derive an even stronger result, in which only the small eigenvalues that are conjugate to the Perron-Frobenius eigenvalue contribute to the complexity of the spectrum. \begin{thm} \label{notfull} Let $b_{PF}$ be the number of large eigenvalues, counted without multiplicity, that are algebraically conjugate to the Perron-Frobenius eigenvalue $\lambda_{PF}$ (including $\lambda_{PF}$ itself), and let $s_{PF}$ be the number of small eigenvalues conjugate to $\lambda_{PF}$. For fixed $\L_f$, the dimension over $\qed$ of the point spectrum tensored with $\qed$ is at most $d(s_{PF}+1)$. \end{thm} \medskip\noindent\textbf{Proof.\enspace} As a first step we diagonalize $M$ over the rationals as far as possible. By rational operations we can always put $M$ in block-diagonal form, where the characteristic polynomial of each block is a power of an irreducible polynomial. Since the Perron-Frobenius eigenvalue $\lambda _{PF}$ has both geometric and algebraic multiplicity $d$, every eigenvalue algebraically conjugate to $\lambda _{PF}$ also has multiplicity $d$. Thus there are $d$ blocks whose characteristic polynomial has $\lambda _{PF}$ for a root. We consider the constraints on the spectrum that can be obtained from these blocks alone. Consider the projection of $\k \cdot \L_f/(2\pi)$ onto the large eigenspaces of the Perron-Frobenius block. Since only $d(b_{PF}+s_{PF})$ components of $\t$ (expressed in the new basis) contribute, this is the projection of $\qed^{d(b_{PF}+s_{PF})}$ onto $\red^{d b_{PF}}$, whose real span is all of $\red^{d b_{PF}}$. Intersected with the $d$-plane defined by a fixed $\L_f$, this gives a vector space of dimension at most $d(s_{PF}+1)$ in which $\k$ can live. \hfill $\square$ \section{One dimensional substitutions revisited} When discussing 1-dimensional substitutions, with $n$ tile types $t_1, \ldots, t_n$, the conventional object of study is the {\em substitution matrix}, whose $(i,j)$ entry gives the number of times that $t_i$ appears in $\sigma(t_j)$. Indeed, in our previous study \cite{CS} of one dimensional tilings, all the results were phrased in terms of eigenvalues and eigenspaces of the substitution matrix, rather than on the induced action of $\sigma$ on homology. These results become much simpler when viewed homologically. In this section, the substitution matrix will be denoted $M_s$, while the matrix that gives the action of $\sigma$ on a basis of recurrences will be denoted $M_h$. In \cite{CS} we defined, for each recurrence $(z_1,z_2)$, a vector in $\zed^n$ that listed how many of each tile type appears in the (unique) path from $z_1$ to $z_2$. This vector $\v$ was called {\em full} if the vectors $(M_s)^k \v$, with $k$ ranging from 0 to $n-1$, were linearly independent. Many of our theorems required the existence of a recurrence with a full vector. This is a strong condition, as it implies that $H_1(\Gamma)$ is a lattice of rank $n$. This is true when $n=2$, or when the characteristic polynomial of $M_s$ is irreducible, but is typically false for more complicated substitutions. When $H_1(\Gamma)$ has rank less than $n$, there are deformations of tile lengths that have no effect on the lengths of recurrences, and so lead to MLD tilings. By looking at $M_h$ rather than $M_s$, we automatically avoid those extraneous modes. Consider the difference between the following theorem, proved in \cite{CS}, and its restatement in terms of $M_h$: \begin{thm}[CS] Suppose that all the eigenvalues of $M_s$ are of magnitude 1 or greater, and that there exists a recurrence with a full vector. If the ratio of any two tile lengths is irrational, then the point spectrum is trivial. \end{thm} \begin{thm}[Corollary of Theorem 4.2] Suppose that all the eigenvalues of $M_h$ are of magnitude 1 or greater. If the ratio of the lengths of any two recurrences is irrational, then the point spectrum is trivial. \end{thm} In addition to $M_h$ not containing irrelevant information found in $M_s$, $M_h$ may contain some relevant information {\em not} found in $M_s$. If $H_1(\Gamma^{(1)})$ has higher rank than $H_1(\Gamma^{(0)})$, then $M_h$ contains information about the dynamical impact of changing the sizes of the collared tiles, and not merely the effect of changing the original, uncollared tiles. As an example, consider the Thue-Morse substitution $(a \to ab, b \to ba)$, in which $M_s=\begin{pmatrix} 1 & 1 \cr 1 & 1 \end{pmatrix}$ has eigenvalues 2 and 0. $\Gamma$ is the wedge of two circles, one representing the tile $a$ and one representing the tile $b$, so $H_1(\Gamma)=\zed^2$, and the action of $\sigma$ on $H_1(\Gamma)$ is described by $M_s$. However, $H_1(\Gamma^{(1)})$ has rank 3 \cite{AP}, and the eigenvalues of $M_h$ are 2, $-1$, and 0. The additional large eigenvalue $-1$ shows that the dynamics of the Thue-Morse tiling space are in fact sensitive to changes in tile size. 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Queff\'{e}lec, Substitution dynamical systems -- spectral %analysis. \emph{Lecture Notes in Mathematics,} \textbf{1294 (}1987), %Springer-Verlag, Berlin. \bibitem[RS]{RS} C. Radin and L. Sadun, Isomorphism of hierarchical structures, \emph{Ergodic Theory \& Dynamical Systems} \textbf{21} (2001), 1239-1248. \bibitem[Sa]{Sa} L.~Sadun, Tiling spaces are inverse limits, preprint 2002, {\tt math.DS/0210179}. \bibitem[So1]{So1} B. Solomyak, Dynamics of self-similar tilings. \emph{Ergodic Theory \& Dynamical Systems} \textbf{17} (1997), 695--738; Corrections in \textbf{19} (1999), 1685. \bibitem[So2]{So2} B. Solomyak, Non-periodicity implies unique composition for self-similar translationally-finite tilings. \emph{Discrete Comput. Geom.} \textbf{20} (1998), 265--279. \bibitem[SW]{SW} L. Sadun and R. F. Williams, Tiling spaces are Cantor set fiber bundles. \emph{Ergodic Theory }$ \emph{\&}$\emph{\ Dynamical Systems} {\bf 23} (2003), 307--316. \end{thebibliography} \end{document} ---------------0306131414466 Content-Type: application/postscript; name="bob1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bob1.ps" %!PS-Adobe-2.0 %%BoundingBox 25 95 580 615 %This will be the mesh of our lattice /M 15 def M 16 mul 47 M mul moveto /Times-Roman findfont 25 scalefont setfont (180 Tiles) show /Tr {/y2 exch def /x2 exch def x2 M mul y2 M mul translate } def .8 setgray 1 5 Tr [] 0 setdash 0 M M 41 mul {0 moveto 0 41 M mul rlineto stroke}for 0 M M 41 mul {0 exch moveto M 41 mul 0 rlineto stroke}for 0 setgray -1 M mul 11 M mul translate %-56 -150 translate %this draws a line from 0 0 to the point /Dl {/y2 exch def /x2 exch def 0 0 moveto x2 M mul y2 M mul lineto stroke } def %This draws a line from the current point to the input point /Pl {/y2 exch def /x2 exch def x2 M mul y2 M mul rlineto } def %10 -5 translate /a{0 0 moveto 1 4 Pl stroke} def /Ta{0 0 moveto -1 4 Pl stroke}def /T2a {0 0 moveto -3 2 Pl stroke}def /T3a {0 0 moveto -4 0 Pl stroke} def /T4a {0 0 moveto -3 -2 Pl stroke} def /T5a{-1 -1 scale a -1 -1 scale}def /T6a{-1 -1 scale Ta -1 -1 scale}def /T7a{-1 -1 scale T2a -1 -1 scale}def /T8a{-1 -1 scale T3a -1 -1 scale}def /T9a{-1 -1 scale T4a -1 -1 scale}def /b{ T6a}def /Tb{ T7a}def /T2b{ T8a}def /T3b{ T9a}def /T4b{ a}def /T5b{ Ta}def /T6b{ T2a}def /T7b{ T3a}def /T8b{ T4a}def /T9b{ T5a}def /c{0 0 moveto 2 0 Pl stroke}def /Tc{0 0 moveto 2 2 Pl stroke}def /T2c{0 0 moveto 1 2 Pl stroke}def /T3c{0 0 moveto -1 2 Pl stroke}def /T4c{0 0 moveto -2 2 Pl stroke}def /T5c{0 0 moveto -2 0 Pl stroke}def /T6c{0 0 moveto -2 -2 Pl stroke}def /T7c{0 0 moveto -1 -2 Pl stroke}def /T8c{0 0 moveto 1 -2 Pl stroke}def /T9c{0 0 moveto 2 -2 Pl stroke}def /d{0 0 moveto 6 0 Pl stroke}def /Td{0 0 moveto 5 4 Pl stroke}def /T2d{0 0 moveto 2 6 Pl stroke}def /T3d{0 0 moveto 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/x2 exch def x2 y2 Tr 0 0 2 0 360 arc fill x2 neg y2 neg Tr} def %the vertices %/Times-Roman findfont 25 scalefont setfont (50 Tiles) show 0 0 Dot 3 2 Dot 4 6 Dot 6 0 Dot 2 6 Dot 7 2 Dot 8 6 Dot 10 0 Dot 3 10 Dot -3 10 Dot 0 8 Dot -1 4 Dot -4 6 Dot -4 0 Dot -5 4 Dot -7 2 Dot -8 6 Dot -7 -2 Dot -8 -6 Dot -10 0 Dot -5 -4 Dot -4 -6 Dot -3 -10 Dot -1 -4 Dot 0 -8 Dot 2 -6 Dot 3 -10 Dot 4 -6 Dot 8 -6 Dot 3 -2 Dot 7 -2 Dot 14 0 Dot 20 0 Dot 17 2 Dot 11 4 Dot 15 4 Dot 14 6 Dot 17 6%here 10 8 Dot 7 10 Dot 13 10 Dot 0 12 Dot 10 12 Dot 4 14 Dot 8 14 Dot 1 16 Dot 7 16 Dot 11 16 Dot 0 20 Dot 6 20 Dot -11 16 Dot -5 16 Dot -1 16 Dot -16 12 Dot -10 12 Dot -4 12 Dot -7 10 Dot -15 8 Dot -11 8 Dot -18 6 Dot -14 6 Dot -15 4 Dot -11 4 Dot -16 0 Dot -12 0 Dot -15 -4 Dot -11 -4 Dot -18 -6 Dot -14 -6 Dot -15 -8 Dot -11 -8 Dot -7 -10 Dot -16 -12 Dot -10 -12 Dot -4 -12 Dot -8 -14 Dot 1 -16 Dot -11 -16 Dot -5 -16 Dot 0 -20 Dot 6 -20 Dot 7 -16 Dot 11 -16 Dot 4 -14 Dot 8 -14 Dot 0 -12 Dot 10 -12 Dot 13 -10 Dot 10 -8 Dot 14 -6 Dot 18 -6 Dot 11 -4 Dot 15 -4 Dot 17 -2 Dot %here is joint 7 -2 Tr -7 2 Tr 14 0 Tr T3a b d T4b -14 0 Tr 17 2 Tr T4a Tb a T4c -17 -2 Tr 11 4 Tr T5a T2b T2d T6b -11 -4 Tr 14 6 Tr T4a T8c T8a T5b -14 -6 Tr 18 6 Dot 18 6 Tr T8b T8d T4d -18 -6 Tr 10 8 Tr T6c T6a T3b T2a -10 -8 Tr 7 10 Tr b d T4b T3a -7 -10 Tr 13 10 Tr -13 -10 Tr 0 12 Tr T7a T4b T4d T8b 0 -12 Tr 10 12 Tr T4a Tb a T4c -10 -12 Tr 4 14 Tr T5a T2b T2d T6b -4 -14 Tr 8 14 Tr -8 -14 Tr 1 16 Tr -1 -16 Tr 7 16 Tr T8c T8a T5b T4a -7 -16 Tr 11 16 Tr T8b T8d T4d -11 -16 Tr 3 18 Dot 3 18 Tr T6c T6a T3b T2a -3 -18 Tr 0 20 Tr T6d b d 0 -20 Tr 6 20 Tr -6 -20 Tr -1 16 Tr c a T7b T6a 1 -16 Tr -11 16 Tr T6d b d 11 -16 Tr -8 14 Dot -8 14 Tr T6c T6a T3b T2a 8 -14 Tr -12 12 Dot -12 12 Tr c a T7b T6a 12 -12 Tr -4 12 Tr T8c T8a T5b T4a 4 -12 Tr -7 10 Tr T2b T2d T6b T5a 7 -10 Tr -15 8 Tr T8c T8a T5b T4a 15 -8 Tr -11 8 Tr T8b T4b T4d T7a 11 -8 Tr -18 6 Tr T8d T2b T2d 18 -6 Tr -8 6 Tr Td T3d T5d T7d Tb 8 -6 Tr -15 4 Tr T2c T2a T9b T8a 15 -4 Tr -11 4 Tr T6d T6b T9a b 11 -4 Tr -12 0 Tr c a T7b T6a 12 0 Tr -15 -4 Tr T8c T8a T5b T4a 15 4 Tr -11 -4 Tr T4b T4d T8b T7a 11 4 Tr -18 -6 Tr T8d T2b T2d 18 6 Tr -15 -8 Tr T2c T2a T9b T8a 15 8 Tr -11 -8 Tr T6b T6d b T9a 11 8 Tr %to here only -7 -10 Tr T2b Ta T8b T8d 7 10 Tr -3 -10 Tr 3 10 Tr -12 -12 Dot -12 -12 Tr c a T7b T6a 12 12 Tr -4 -12 Tr T2c T2a T9b T8a 4 12 Tr 0 -12 Tr T9a T6b T6d b 0 12 Tr -8 -14 Tr T4c T4a Tb a 8 14 Tr -1 -16 Tr c a T7b T6a 1 16 Tr -1 -16 Dot -11 -16 Tr d T4b T4d 11 16 Tr -5 -16 Tr 5 16 Tr -1 -16 Tr 1 16 Tr 3 -18 Tr a T4c T4a Tb -3 18 Tr 0 -20 Tr T4d T4b d 0 20 Tr 7 -16 Tr T9b T8a T2c T2a -7 16 Tr 11 -16 Tr T2d T6b T6d -11 16 Tr 4 -14 Tr T2b Ta T8b T8d -4 14 Tr 8 -14 Tr -8 14 Tr 0 -12 Tr 0 12 Tr 10 -12 Tr T6c T6a T3b T2a -10 12 Tr 3 -10 Tr T5d T7d T9d Td T5b -3 10 Tr 7 -10 Dot 7 -10 Tr b d T4b T3a -7 10 Tr 10 -8 Tr T4a Tb T4c a -10 8 Tr 14 -6 Tr T2c T2a T9b T8a -14 6 Tr 18 -6 Tr T6d T6b T2d -18 6 Tr 11 -4 Tr T8d T2b Ta T8b -11 4 Tr 15 -4 Tr -15 4 Tr 17 -2 Tr T6c T6a T3b T2a -17 2 Tr showpage ---------------0306131414466 Content-Type: application/postscript; name="chairsubs.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="chairsubs.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Title: /tmp/xfig-fig010840 %%Creator: fig2dev %%CreationDate: Tue Nov 19 15:22:56 1996 %%For: radin@marie (Charles Radin,dept,fac,000000) %%BoundingBox: 0 0 331 115 %%Pages: 0 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} 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