Shifts and Subshifts over $\mathbb {Z}$

Abstract

In this chapter, we introduce the shift map $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ on the space of bi-infinite sequences of points in a topological space K.

Appendices

Notes

The branch of the theory of dynamical systems devoted to the investigation of the dynamical properties of shift maps $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ is known as symbolic dynamics. The most studied case is when the symbol space K is a finite or countably infinite discrete space (see for example [59, 70]).

Theorem 7.2.1 is in [44, 74]. Theorem 7.3.5 is a particular case of Proposition 1.9.A in [44].

As mentioned in the Notes on Chap. 6, Boltyanskiǐ [15, 16] gave examples of compact metrizable spaces K satisfying ${{\mathrm{stabdim}}}(K) < \dim (K)$. For such spaces K, we have that ${{\mathrm{mdim}}}(K^\mathbb {Z},\sigma ) < \dim (K)$ by Theorem 7.1.3. It would be interesting to find an example of a compact metrizable space K for which ${{\mathrm{mdim}}}(K^\mathbb {Z},\sigma ) < {{\mathrm{stabdim}}}(K)$.

Exercises

 

• 7.1 Let K be a metrizable space and let $d_K$ be a bounded metric on K that is compatible with its topology. Show that if the set $K^\mathbb {Z}$ is equipped with the metric d defined by Formula (7.1.1), then the shift map $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ is 2-Lipschitz, i.e., it satisfies $d(\sigma (x),\sigma (y)) \le 2d(x,y)$ for all $x,y \in K^\mathbb {Z}$.

• 7.2 Let X be a topological space and $T :X \rightarrow X$ a continuous map. The dynamical system (X, T) is called topologically mixing if, given any two non-empty open subsets U and V of X, there are only finitely many $n \in \mathbb {Z}$ such that $T^n(U) \cap V = \varnothing$. Let K be a topological space and let $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ denote the shift map on $K^\mathbb {Z}$. Show that the dynamical system $(K^\mathbb {Z},\sigma )$ is topologically mixing.

• 7.3 Let K and L be topological spaces. Show that the dynamical systems $((K \times L)^\mathbb {Z}, \sigma _{K \times L})$ and $(K^\mathbb {Z}\times L^\mathbb {Z}, \sigma _K \times \sigma _L)$ are topologically conjugate.

• 7.4 Let $K := \{0,1\}$ and $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ the shift map. Let X denote the subset of $K^\mathbb {Z}$ consisting of all $x = (x_i)_{i \in \mathbb {Z}} \in K^\mathbb {Z}$ such that there is at most one integer $i \in \mathbb {Z}$ with $x_i = 1$.

  • (a) Show that X is a subshift of $K^\mathbb {Z}$.

  • (b) Show that the subshift X is not of finite type.

  • (c) Show that the dynamical system $(X,\sigma )$ is not topologically mixing.

• 7.5 Let $K := \{0,1\}$ and $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ the shift map. Let X denote the subset of $K^\mathbb {Z}$ consisting of all sequences with no consecutive 1s. Let Y denote the subset of $K^\mathbb {Z}$ consisting of all sequences such that between any two 1s there are always an even number of 0s.

  • (a) Show that X and Y are subshifts of $K^\mathbb {Z}$. The subshifts X is called the golden mean subshift and the subshift Y is called the even subshift .

  • (b) Show that the dynamical systems $(X,\sigma )$ and $(Y,\sigma )$ are not topologically conjugate. Hint: count fixed points.

  • (c) Show that the subshift X is of finite type but Y is not.

  • (d) Show that the dynamical systems $(X,\sigma )$ and $(Y, \sigma )$ are both topologically mixing.

  • (e) Show that the map $f :X \rightarrow Y$ that sends each $x = (x_i)_{i \in \mathbb {Z}} \in X$ to the sequence $y = (y_i)_{i \in \mathbb {Z}}$ given by

    $$ y_i = {\left\{ \begin{array}{ll} 1 &{}\text {if } x_i = x_{i + 1} = 0 \\ 0&{} \text {otherwise}, \end{array}\right. } $$

    is well defined, continuous and surjective, ant that it commutes with the shift.

• 7.6 Let K be a finite discrete topological space with cardinality k. Given an integer $n \ge 1$, let $\pi _n :K^\mathbb {Z}\rightarrow K^n$ be the map defined by

  $$ \pi _n(x) := (x_0,x_1,\dots ,x_{n - 1}) $$

  for all $x = (x_i)_{i \in \mathbb {Z}} \in K^\mathbb {Z}$. Let $X \subset K^\mathbb {Z}$ be a non-empty subshift. For $n \ge 1$, let $\gamma _n(X)$ denote the cardinality of the set $\pi _n(X)$.

  • (a) Show that the sequence $(\log \gamma _n(X))_{n \ge 1}$ is subadditive.

  • (b) Show that the limit

    $$ h(X) := \lim _{n \rightarrow \infty } \frac{\log \gamma _n(X)}{n} $$

    exists and satisfies $0 \le h(X) \le \log k$. This limit is called the entropy of the subshift X.

  • (c) Show that $h(X) = h_{top}(X,\sigma )$, where $h_{top}(X,\sigma )$ is the topological entropy of the dynamical system $(X,\sigma )$ (cf. Exercise 6.11).

  • (d) Take $K = \{0,1\}$ and suppose that $X \subset K^\mathbb {Z}$ is the subshift considered in Exercise 7.4. Show that $h(X) = 0$.

  • (e) Take again $K = \{0,1\}$. Show that the golden mean subshift X and the even subshift Y (cf. Exercise 7.5) have the same entropy $h(X) = h(Y) = \log \phi$, where $\phi := (1 + \sqrt{5})/2$ is the golden mean. Hint: check that $\gamma _{n + 2}(X) = \gamma _{n + 1}(X) + \gamma _n(X)$ and $\gamma _n(Y) = \gamma _{n + 1}(X) - 1$ for all $n \ge 1$.

  • (f) Show that if $Y \subset K^\mathbb {Z}$ is a subshift such that $X \subset Y$, then $h(X) \le h(Y)$.

  • (g) Show that if X is the subshift of block type associated with a pair (q, B), where $q \ge 1$ is an integer and $B \subset K^q$ has cardinality b, then

    $$ h(X) = \frac{\log b}{q \log k}. $$

  • (h) Show that if $\rho$ is a real number such that $0 \le \rho \le \log k$, then there exists a subshift $X \subset K^\mathbb {Z}$ such that $h(X) = \rho$. Hint: adapt the ideas used in the proof of Theorem 7.6.2.

• 7.7 Let X be a compact metrizable space and $T :X \rightarrow X$ a homeomorphism. Let d be a metric on X defining the topology. One says that the homeomorphism T is expansive if there exists a real number $\delta > 0$ satisfying the following condition: if two points $x, y \in X$ are such that $d(T^n(x),T^n(y)) \le \delta$ for all $n \in \mathbb {Z}$ then $x = y$.

  • (a) Show that the above definition does not depend on the choice of d.

  • (b) Show that the following conditions are equivalent: (1) the homeomorphism T is expansive and $\dim (X) = 0$; (2) there exist a finite discrete topological space K and a subshift $Y \subset K^\mathbb {Z}$ such that the dynamical systems (X, T) and $(Y,\sigma )$ are topologically conjugate.

• 7.8 Let K be a finite discrete topological space and let $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ denote the shift map. One says that a map $f :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ is a cellular automaton if f is continuous and satisfies $f \circ \sigma = \sigma \circ f$.

  • (a) Let $f :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ be a map. Show that f is a cellular automaton if and only if it satisfies the following condition: there exist an integer $n \ge 0$ and a map $\mu :K^{2n + 1} \rightarrow K$ such that, for all $x = (x_i)_{i \in \mathbb {Z}} \in K^\mathbb {Z}$, one has $f(x) = (y_i)_{i \in \mathbb {Z}}$, where

    $$ y_i = \mu (x_{i - n}, x_{i - n + 1}, \dots ,x_{i + n}) $$

    for all $i \in \mathbb {Z}$.

  • (b) Show that if $f :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ and $g :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ are cellular automata, then their composite map $f \circ g :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ is also a cellular automaton.

  • (c) Show that if $f :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ is a bijective cellular automaton, then its inverse map $f^{-1} :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ is also a cellular automaton.

• 7.9 Let $K := [0,1]$ denote the unit segment and $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ the shift map. Let X denote the subset of $K^\mathbb {Z}$ consisting of all $x = (x_i)_{i \in \mathbb {Z}} \in K^\mathbb {Z}$ such that $x_i + x_{i + 1} = 1$ for all $i \in \mathbb {Z}$.

  • (a) Show that X is a subshift of finite type of $K^\mathbb {Z}$.

  • (b) Show that $(X,\sigma )$ is topologically conjugate to the dynamical system (K, f), where $f :K \rightarrow K$ is the map defined by $f(t) = 1 - t$ for all $t \in K$.

  • (c) Show that ${{\mathrm{mdim}}}(X,\sigma ) = 0$.

• 7.10 Let $K \subset \mathbb {C}$ denote the unit circle $K := \mathbb {S}^1 = \{z \in \mathbb {C}: |z| = 1\}$ and let $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ be the shift map. Consider the subset $X \subset K^\mathbb {Z}$ defined by

  $$ X := \{ (z_n)_{n \in \mathbb {Z}} \in K^\mathbb {Z}: z_n^2 = z_{n + 1}^3 \text { for all } n \in \mathbb {Z}\}. $$

  • (a) Show that X is a subshift of finite type.

  • (b) Show that ${{\mathrm{mdim}}}(X,\sigma ) = 0$. Hint: observe that $w_n(X)$, where $w_n :K^\mathbb {Z}\rightarrow K^n$ is defined by (7.3.1), is homeomorphic to $\mathbb {S}^1$ for all $n \ge 1$ and then apply Theorem 7.3.5.

• 7.11 Let K be a compact Hausdorff space and let $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ denote the shift map. Show that a closed subset $X \subset K^\mathbb {Z}$ is a subshift of finite type if and only if it satisfies the following condition: there exists a finite subset $\Omega \subset \mathbb {Z}$ such that

  $$ X = \{ x \in K^\mathbb {Z}\mid \; \pi _\Omega (\sigma ^n(x)) \in \pi _\Omega (X) \text { for all } n \in \mathbb {Z}\} $$

  (here $\pi _\Omega :K^\mathbb {Z}\rightarrow K^\Omega$ denotes the canonical projection map).