Applications of Mean Dimension to Embedding Problems

Abstract

In this chapter, we prove the embedding theorem of Jaworski (Theorem 8.3.1) which asserts that every dynamical system (X, T), where T is a homeomorphism without periodic points of a compact metrizable space X such that $\dim (X) < \infty$, embeds in the shift $(\mathbb {R}^\mathbb {Z},\sigma )$.

Appendices

Notes

Theorem 8.3.1 was obtained by Jaworski in his Ph.D. thesis [51, Th. IV.1] (see [11, p. 194]). A homeomorphism T of a topological space X generates a continuous action of the additive group $\mathbb {Z}$ on X, given by the map $\mathbb {Z}\times X \rightarrow X$ that sends each pair $(n,x) \in \mathbb {Z}\times X$ to the point $T^n(x)$. A continuous action of the additive group $\mathbb {R}$ is called a flow. More precisely, a flow on a topological space X consists of a family $\varphi = (\varphi _t)_{t \in \mathbb {R}}$ of homeomorphisms of X such that the map $(t,x) \mapsto \varphi _t(x)$ is continuous and $\varphi _t \circ \varphi _s = \varphi _{t + s}$ for all $t,s \in \mathbb {R}$. A point $x \in X$ is a fixed point of the flow $(X,\varphi )$ if $\varphi _t(x) = x$ for all $t \in \mathbb {R}$. One says that the flow $(X,\varphi )$ embeds in the flow $(Y,\psi )$ if there exists a topological embedding such that $h \circ \varphi _t = \psi _t \circ h$ for all $t \in \mathbb {R}$. Denote by $C(\mathbb {R})$ the set consisting of all continuous maps $f :\mathbb {R}\rightarrow \mathbb {R}$ and equip $C(\mathbb {R})$ with the topology of uniform convergence on compact subsets of $\mathbb {R}$. Consider the flow $\lambda = (\lambda _t)_{t \in \mathbb {R}}$ on $C(\mathbb {R})$ defined by $\lambda _t(f)(u) := f(u + t)$ for all $f \in C(\mathbb {R})$ and $t,u \in \mathbb {R}$. The flow $(C(\mathbb {R}),\lambda )$ is a continuous version of the shift $(\mathbb {R}^\mathbb {Z},\sigma )$. Theorem 8.3.1 is analogous to a theorem of Bebutoff [13] (see also [53, 55], [11, p. 184]) which asserts that every flow without fixed points $(X,\varphi )$, where X is a compact metrizable space, embeds in the flow $(C(\mathbb {R}),\lambda )$. Note however that, in contrast with Theorem 8.3.1, there is no hypothesis about the topological dimension of X in the statement of Bebutoff’s theorem. The set of fixed points of the flow $(C(\mathbb {R}),\lambda )$ is the set of constant functions and is therefore homeomorphic to the real line $\mathbb {R}$. It follows that a necessary condition for a flow $(X,\varphi )$ to be embeddable in the flow $(C(\mathbb {R}),\lambda )$ is that the set of fixed points of $(X,\varphi )$ is homeomorphic to a subset of $\mathbb {R}$. By a result of Kakutani [53], which extends Bebutoff’s theorem, it turns out that this condition is also sufficient for flows on compact metrizable spaces.

The construction of the counterexamples described in Sect. 8.4 is due to Lindenstrauss and Weiss (see [74, Proposition 3.5]).

Let T be a homeomorphism of a compact metrizable space X. In [72, Th. 5.1], Lindenstrauss proved that if (X, T) is minimal and ${{\mathrm{mdim}}}(X,T) < d/36$ for some integer $d \ge 1$, then the dynamical system (X, T) can be embedded in the shift $((\mathbb {R}^d)^\mathbb {Z},\sigma )$. On the other hand, Lindenstrauss and Tsukamoto [73] constructed, for any integer $d \ge 1$, a compact metrizable space X admitting a homeomorphism T such that the dynamical system (X, T) is minimal and satisfies ${{\mathrm{mdim}}}(X,T) = d/2$ but cannot be embedded in the shift $((\mathbb {R}^d)^\mathbb {Z},\sigma )$. For additional results related to Jaworski’s theorem and the question of the embeddability of dynamical systems in the shift on $(\mathbb {R}^d)^\mathbb {Z}$, see also [45, 46].

Exercises

 

• 8.1 Let K be a topological space and let $\sigma$ denote the shift map on $K^\mathbb {Z}$. Show that the set of periodic points of $\sigma$ is dense in $K^\mathbb {Z}$.

• 8.2 Let K be an accessible separable space with more than one point. Show that the shift $(K^\mathbb {Z},\sigma )$ is topologically transitive but not minimal.

• 8.3 Let K be a topological space and let $\sigma :K^\mathbb {Z}\rightarrow K^\mathbb {Z}$ denote the shift map on $K^\mathbb {Z}$. Let q be a positive integer and B a closed subset of $K^q$. Let $X \subset K^\mathbb {Z}$ denote the subshift of block-type associated with (q, B). Show that the set of periodic points of the dynamical system $(X,\sigma )$ is dense in X.

• 8.4 ( Adding machines ). Let $(a_n)_{n \in \mathbb {N}}$ be a sequence of positive integers. Consider the product space

  $$ X := \prod _{n \in \mathbb {N}} \{0,1,\dots ,a_n -1 \} $$

  and the map $T :X \rightarrow X$ defined in the following way. If $x =(x_n)_{n \in \mathbb {N}} \in X$ with $x_n = a_n - 1$ for all $n \in \mathbb {N}$ then we take $T(x) := (y_n)_{n \in \mathbb {N}}$, where $y_n = 0$ for all $n \in \mathbb {N}$. Otherwise, there is a largest integer $n_0 \in \mathbb {N}$ such that $x_n = a_n - 1$ for all $n \le n_0$, and we take $T(x) := (y_n)_{n \in \mathbb {N}}$, where $y_n = 0$ for all $n \le n_0$, $y_{n_0 + 1} = x_{n_0 + 1} + 1$, and $y_n = x_n$ for all $n \ge n_0 + 2$.

  • (a) Show that T is a homeomorphism.

  • (b) Show that the dynamical system (X, T) is minimal.

• 8.5 Let X be a topological space and $T :X \rightarrow X$ a homeomorphism. Show that the dynamical system (X, T) is minimal if and only if every non-empty open subset $U \subset X$ satisfies $\bigcup _{n \in \mathbb {Z}} T^n(U) = X$.

• 8.6 Let X be a compact space and $T :X \rightarrow X$ a homeomorphism. Show that the dynamical system (X, T) is minimal if and only if, for every non-empty open subset $U \subset X$, there exists $n \in \mathbb {N}$ such that $\bigcup _{k = -n}^n T^k(U) = X$.

• 8.7 Let X be a non-empty compact Hausdorff space and $T :X \rightarrow X$ a homeomorphism. Show that there exists a non-empty closed subset $Y \subset X$ with $T(Y) = Y$ such that the dynamical system (Y, T) is minimal. Hint: use Zorn’s lemma.

• 8.8 Show that every closed subgroup G of $\mathbb {R}$ such that $\{0\} \not = G \not = \mathbb {R}$ is infinite cyclic. Deduce that if $T :\mathbb {S}^1 \rightarrow \mathbb {S}^1$ is a rotation of angle $\theta$ with $\theta /\pi$ irrational, then the dynamical system $(\mathbb {S}^1,T)$ is minimal.

• 8.9 Let S be a subset of $\mathbb {Z}$. One says that S is syndetic if there exists a finite subset $F \subset \mathbb {Z}$ such that $S + F = \mathbb {Z}$. Show that the following conditions are equivalent: (1) S is syndetic; (2) there exists an integer $N \ge 1$ such that S is N-dense in $\mathbb {Z}$; (3) there exists an integer $k \ge 1$ such that one has $\{i, i\,+\,1,\dots ,i\,+\,k\,-\,1\}\,\cap \,S \not = \varnothing$ for all $i \in \mathbb {Z}$; (4) S has bounded gaps, i.e., there exists an integer $L \ge 1$ such that every subset of $\mathbb {Z}{\setminus } S$ consisting of consecutive integers has cardinality at most L.

• 8.10 Let X be a topological space equipped with a homeomorphism $T :X \rightarrow X$. A point $x \in X$ is called almost-periodic if, for every neighborhood U of x, the set consisting of all $n \in \mathbb {Z}$ such that $T^n(x) \in U$ is a syndetic subset of $\mathbb {Z}$ (see Exercise 8.9).

  • (a) Show that every periodic point of T is almost-periodic.

  • (b) Show that if X is compact and (X, T) is minimal then every point $x \in X$ is almost-periodic.

  • (c) Suppose that X is a compact Hausdorff space and $x \in X$ is almost-periodic. Let Y denote the closure in X of the orbit of x. Show that the dynamical system (Y, T) is minimal.

• 8.11 Let $n,m \in \mathbb {Z}$. Show that the shift $((\mathbb {R}^n)^\mathbb {Z},\sigma )$ embeds in the shift $((\mathbb {R}^m)^\mathbb {Z},\sigma )$ if and only if $n \le m$.

• 8.12 Let X be a compact metrizable space and $T :X \rightarrow X$ a homeomorphism. Let d be a metric on X compatible with the topology. The dynamical system (X, T) is called distal if the following condition is satisfied: given any pair of distinct points x and y in X, there exists a real number $\varepsilon > 0$ such that $d(T^n(x),T^n(y)) \ge \varepsilon$ for all $n \in \mathbb {Z}$.

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

  • (b) Show that if the dynamical system (X, T) is both distal and minimal, then it embeds in the shift $(\mathbb {R}^\mathbb {Z},\sigma )$.

• 8.13 Let $\mathbb {S}^1 := \{(x_1,x_2) \in \mathbb {R}^2 \mid \; x_1^2 + x_2^2 = 1\}$ denote the unit circle in $\mathbb {R}^2$. Let $T :\mathbb {S}^1 \rightarrow \mathbb {S}^1$ be the half-turn given by $T(x_1,x_2) = -(x_1,x_2)$. Show that the dynamical system $(\mathbb {S}^1,T)$ does not embed in the shift $(\mathbb {R}^\mathbb {Z},\sigma )$. Hint: observe that the dynamical system $({{\mathrm{Per}}}_2(T),T)$ does not embed in the dynamical system $({{\mathrm{Per}}}_2(\sigma ),\sigma )$.

• 8.14 (cf. [51, Example 4.1]). Let Y be a compact metrizable space and $n \ge 2$ an integer. Consider the product space $X := Y \times \{0,1,\dots ,n - 1\}$ and the homeomorphism $T :X \rightarrow X$ defined, for all $x = (y,k) \in X$, by

  $$ T(x) := {\left\{ \begin{array}{ll} (y,k + 1) &{}\text { if } k \le n - 2 \\ (y,0) &{}\text { if } k = n - 1. \end{array}\right. } $$

  Show that the dynamical system (X, T) embeds in the shift $(\mathbb {R}^\mathbb {Z},\sigma )$ if and only if Y is topologically embeddable in $\mathbb {R}^n$.

• 8.15 Let X be a compact metrizable space with $\dim (X) = n < \infty$ and $T :X \rightarrow X$ a homeomorphism. Suppose that every orbit of T contains at least $6n + 1$ distinct points. Show that the dynamical system (X, T) embeds in the shift $(\mathbb {R}^\mathbb {Z},\sigma )$. Hint: observe that Lemma 8.3.3 remains valid for $m = 2n + 1$.