Mean Topological Dimension for Continuous Maps

Abstract

In this chapter, the term “dynamical system” refers to a pair (X,T), where X is a topological space and T a continuous map from X into itself.

Appendices

Notes

Subadditivity plays an important role in many branches of pure and applied mathematics. Fekete’s lemma (Proposition 6.2.3), which is named after Fekete (cf. [35, Satz 2] and also [90, p. 198]) has been generalized in various directions. For example, it is known [63, Theorem 16.2.9] that if f is a measurable subadditive real-valued map on $\mathbb {R}^n$ then, for every $x \in \mathbb {R}^n$, the function $g(t) := f(tx)/t$ admits $\inf _{t > 0} g(t) \in \mathbb {R}\cup \{-\infty \}$ as a limit as t tends to infinity.

There exist compact metrizable spaces X for which Inequality (6.2.3) is strict. Actually, Boltyanskiǐ [15, 16] gave examples of compact metrizable spaces X satisfying $\dim (X) = 2$ and $\dim (X \times X) = 3$. For such a space X, we have that ${{\mathrm{stabdim}}}(X) \le 3/2 < \dim (X)$. Since the inequality $\dim (X\,\times \,Y) \le \dim (X)\,+\,\dim (Y)$ remains valid whenever X and Y are compact Hausdorff or metrizable (see the Notes on Chap. 4), the limit $\lim _{n \rightarrow \infty } \dfrac{\dim (X^n)}{n}$ exists and thus the definition of ${{\mathrm{stabdim}}}(X)$ may be extended to the case when X is compact Hausdorff or metrizable.

Mean topological dimension was introduced by Gromov [44] for studying dynamical properties of certain spaces of holomorphic maps and complex varieties. It was used by Lindenstrauss and Weiss [74] to answer in the negative a question that had been raised by Auslander (see Chap. 8). The paper by Lindenstrauss and Weiss contains many other important results about mean topological dimension. It is shown in particular in [74] that if T is a homeomorphism of a compact metrizable space X such that (X, T) is uniquely ergodic or has finite topological entropy, then ${{\mathrm{mdim}}}(X,T) = 0$ (cf. Exercise 6.11 for the definition of topological entropy).

Exercises

 

• 6.1 Let a be a real number such that $0 \le a \le 1$. Show that the sequence $(n^a)_{n \ge 1}$ is subadditive.

• 6.2 Let C be a positive real number. Show that the sequence $(\log (C + n))_{n \ge 1}$ is subadditive.

• 6.3 Let $T :X \rightarrow X$ be a map from a set X into itself and let F be a finite subset of X. For each integer $n \ge 1$, let $u_n$ denote the cardinality of the set

  $$ F \cup T(F) \cup T^2(F) \cup \dots \cup T^{n - 1}(F) = \bigcup _{k = 0}^{n - 1} T^k(F). $$

  • (a) Show that the sequence $(u_n)_{n \ge 1}$ is subadditive.

  • (b) Show that the sequence $(v_n)_{n \ge 1}$, defined by $v_n := u_{n + 1} - u_n$ for all $n \ge 1$, is non-increasing.

  • (c) Show that there exist integers $\alpha \ge 0$ and $n_0 \ge 1$ such that $v_n = \alpha$ for all $n \ge n_0$.

  • (d) Show that $\lim _{n \rightarrow \infty } \dfrac{u_n}{n} = \alpha$.

• 6.4 Let S be a semigroup , i.e., a set equipped with an associative binary operation. Let A be a non-empty subset of S. For $n \ge 1$, denote by $\gamma _n$ the number of elements $s \in S$ that can be written in the form $s = a_1 a_2 \dots a_k$ with $k \le n$ and $a_i \in A$ for all $1 \le i \le k$. Show that the sequence $(u_n)_{n \ge 1}$ defined by $u_n := \log \gamma _n$ is subadditive.

• 6.5 Let $(u_n)_{n \ge 1}$ be a subadditive sequence of real numbers. Show that if the sequence $(\dfrac{u_n}{n})$ is not convergent then one has $\lim _{n \rightarrow \infty } \dfrac{u_n}{n} = - \infty$.

• 6.6 Let $(u_n)_{n \ge 1}$ be a sequence of real numbers. Suppose that there exists a real number C such that

  $$ u_{n + m} \le u_n + u_m + C $$

  for all $n, m \ge 1$. Show that the sequence $(u_n + C)$ is subadditive. Deduce that the sequence $(\dfrac{u_n}{n})$ either is convergent or has $-\infty$ as a limit.

• 6.7 ( Translation number ). Let $f :\mathbb {R}\rightarrow \mathbb {R}$ be a homeomorphism such that $f(x + k) = f(x) + k$ for all $x \in \mathbb {R}$ and $k \in \mathbb {Z}$. Let $x \in \mathbb {R}$. Show that the limit

  $$ \tau (f) := \lim _{n \rightarrow \infty } \dfrac{f^n(x)}{n} $$

  exists and is finite and that this limit does not depend on the choice of the point $x \in \mathbb {R}$. Hint: first observe that f is increasing and then prove that the sequence $(f^n(0) + 1)_{n \ge 1}$ is subadditive.

• 6.8 Let G be a group. A map $q :G \rightarrow \mathbb {R}$ is called a quasi-homomorphism if the map $(g_1,g_2) \mapsto q(g_1g_2)\,-\,q(g_1)\,-\,q(g_2)$ is bounded on $G \times G$. Let $q :G \rightarrow \mathbb {R}$ be a quasi-homomorphism.

  • (a) Let $g \in G$. Show that the sequence $(\dfrac{q(g^n)}{n})$ is convergent.

  • (b) Consider the map $q_\infty :G \rightarrow \mathbb {R}$ defined by

    $$ q_\infty (g) := \lim _{n \rightarrow \infty } \frac{q(g^n)}{n}. $$

    Show that $q_\infty$ is a quasi-homomorphism.

  • (c) Show that $q_\infty (g^n) = n q_\infty (g)$ and $q_\infty (h^{-1}gh) = q_\infty (g)$ for all $n \in \mathbb {Z}$ and $g,h \in G$.

• 6.9 A map $f :[0,\infty ) \rightarrow \mathbb {R}$ is called subadditive if it satisfies $f(x + y) \le f(x) + f(y)$ for all $x, y \in [0,\infty )$.

  • (a) Let $f :[0,\infty ) \rightarrow \mathbb {R}$ be a continuous subadditive map. By adapting the proof of Proposition 6.2.3, show that the map $g :(0,\infty ) \rightarrow \mathbb {R}$ defined by $g(x) := \dfrac{f(x)}{x}$ has a limit $\lambda \in \mathbb {R}\cup \{-\infty \}$ as $x \rightarrow \infty$ and that one has $\lambda = \inf _{x > 0} g(x)$.

  • (b) Recall that a map $f :[0,\infty ) \rightarrow \mathbb {R}$ is called concave if it satisfies $f((1-t)x + ty) \ge (1-t)f(x) + t f(y)$ for all $x,y \in [0,\infty )$ and $t \in [0,1]$. Show that a concave map $f :[0,\infty ) \rightarrow \mathbb {R}$ is subadditive if and only if $f(0) \ge 0$.

  • (c) Show that there exists a non-linear map $f :\mathbb {R}\rightarrow \mathbb {R}$ such that $f(x + y) = f(x) + f(y)$ for all $x,y \in \mathbb {R}$. Hint: use the fact that $\mathbb {R}$, viewed as a vector space over the field $\mathbb {Q}$, admits a base.

  • (d) Show that if f is as in the previous question then $\dfrac{f(x)}{x}$ has no limit as $x \rightarrow \infty$. (This shows that the hypothesis that f is continuous cannot be removed in the first question.)

• 6.10 Let X be a normal space and $T :X \rightarrow X$ a continuous map. Let $\alpha$ and $\beta$ be finite open covers of X. Show that one has $D(\alpha \vee \beta ,T) \le D(\alpha ,T) + D(\beta ,T)$.

• 6.11 Let X be a non-empty topological space and $T :X \rightarrow X$ a continuous map. Given a finite open cover $\alpha = (U_i)_{i \in I}$ of X, denote by $N(\alpha )$ the smallest integer $k \ge 1$ such that there exists a subset $I_0 \subset I$ with cardinality k satisfying $\bigcup _{i \in I_0}U_i = X$.

  • (a) Let $\alpha$ and $\beta$ be finite open covers of X. Show that $N(\alpha \vee \beta ) \le N(\alpha ) N(\beta )$.

  • (b) Let $\alpha$ be a finite open cover of X and $f :X \rightarrow X$ a continuous map. Show that $N(f^{-1}(\alpha )) \le N(\alpha )$.

  • (c) Let $\alpha$ be a finite open cover of X. Given an integer $n \ge 1$, let

    $$ \omega (\alpha ,T,n) := \bigvee _{k=0}^{n-1} T^{-k}(\alpha ). $$

    Show that the limit

    $$ H_{top}(X,T,\alpha ) := \lim _{n \rightarrow \infty } \frac{\log N( \omega (\alpha ,T,n))}{n} $$

    exists and is finite. Hint: observe that the sequence $(\log N( \omega (\alpha ,T,n)))_{n \ge 1}$ is subadditive. The quantity

    $$ h_{top}(X,T) := \sup _{\alpha } H_{top}(X,T,\alpha ), $$

    where $\alpha$ runs over all finite open covers of X, is called the topological entropy of the dynamical system (X, T).

  • (d) Let Y be a topological space and $S :Y \rightarrow Y$ a continuous map. Suppose that there exists a surjective continuous map $f :Y \rightarrow X$ such that $f \circ S = T \circ f$. Show that one has $h_{top}(X,T) \le h_{top}(Y,S)$.

• 6.12 Let X be a normal space and $T :X \rightarrow X$ a constant map. Show that ${{\mathrm{mdim}}}(X,T) = 0$.

• 6.13 Let (X, d) be a compact metric space and $T :X \rightarrow X$ a map satisfying $d(T(x),T(y)) \le d(x,y)$ for all $x,y \in X$. Show that ${{\mathrm{mdim}}}(X,T) = 0$.

• 6.14 Let $X_1$ and $X_2$ be compact metrizable spaces. Let $T_1 :X_1 \rightarrow X_1$ and $T_2 :X_2 \rightarrow X_2$ be continuous maps. Consider the product map

  $$ T_1 \times T_2 :X_1 \times X_2 \rightarrow X_1 \times X_2 $$

  defined by $T_1 \times T_2 (x_1,x_2) := (T_1(x_1),T_2(x_2))$. Show that

  $$ {{\mathrm{mdim}}}(X_1 \times X_2,T_1 \times T_2) \le {{\mathrm{mdim}}}(X_1,T_1) + {{\mathrm{mdim}}}(X_2,T_2). $$

  Hint: use Theorem 6.5.4 and the result of Exercise 4.7.

• 6.15 Let X and Y be compact metrizable spaces. Let $T :X \rightarrow X$ be a continuous map. Show that

  $$ {{\mathrm{mdim}}}(X\times Y ,T \times {{\mathrm{Id}}}_Y) = {{\mathrm{mdim}}}(X,T). $$

• 6.16 Let (X, d) be a compact metric space and let $T :X \rightarrow X$ be a continuous map. For $n \ge 1$, let $d_n$ be the metric on X defined by $d_n(x,y) := \max _{0 \le k \le n - 1} d(T^k(x),T^k(y))$.

  • (a) (cf. Exercises 4.11 and 3.11). Let $\varepsilon > 0$. Show that the sequence

    $$ ({{\mathrm{Widim}}}_\varepsilon (X,d_n))_{n \ge 1} $$

    is subadditive.

  • (b) Show that the limit

    $$ {{\mathrm{mWidim}}}_\varepsilon (X,d,T) := \lim _{n \rightarrow \infty } \frac{{{\mathrm{Widim}}}_\varepsilon (X,d_n)}{n} $$

    exists and is finite.

  • (c) Show that

    $$ {{\mathrm{mdim}}}(X,T) = \lim _{\varepsilon \rightarrow 0} {{\mathrm{mWidim}}}_\varepsilon (X,d,T). $$