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Let $X$ be a compact metric space and $f:X \to X$ a continuous map. We say that $(X,f)$ is approximated from below by a sequence of compact metric spaces $(X_i)_{i \geq 1}$ and a sequence of continuous transformations $(f_i)_{i \geq 1}$ on $X_i$ if we have:

  • $X_i \subset X_{i+1}$ for every $i \geq 1$;
  • $X = \overline{\mathop{\bigcup}\limits_{i=1}^{\infty}X_i}$;
  • $f_{i+1}|_{X_i} = f_i$, $f|_{X_i} = f_i$ for every $i \geq 1$.

And we say that $(X,f)$ is approximated from above by a sequence of compact metric spaces $(Y_i)_{i \geq 1}$ and a sequence of continuous transformations $(g_i)_{i \geq 1}$ on $Y_i$ if we have:

  • $Y_{i+1} \subset Y_i$ for every $i \geq 1$;
  • $X = \mathop{\bigcap}\limits_{i=1}^{\infty}Y_i$;
  • $g_{i}\mid_{Y_{i+1}} = g_{i+1}$, $f\mid_{Y_i} = g_i$ for every $i \geq 1$.

Assume now that $X$ is approximated from above by a sequence $(X_i,f_i)_{i \geq 1}$ and from below by $(Y_i, g_i)_{i \geq 1}$ and $(X_i,f_i)$ and $(Y_i, g_i)$ are intrinsically ergodic systems for all $i \geq 1$, i.e., each of them has a unique measure of maximal entropy.


  1. Is $(X,f)$ intrinsically ergodic?
  2. Is there any example of a dynamical system approximated from above by intrinsically ergodic systems that is not intrinsically ergodic? In February 2012 I asked a question about limits on intrinsically ergodic systems (see link attached) Limits of intrinsically ergodic systems, the answer gave me a counterexample for spaces approximated from below.

(If it helps, each $(X_i,f_i)$ and each $(Y_i, g_i)$ in my set-up is an intrinsically ergodic subshift of finite type for every $i$).

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1 Answer

up vote 6 down vote accepted

Every topologically transitive shift space, whether intrinsically ergodic or not, can be approximated from above by intrinsically ergodic systems.

Indeed, given a finite alphabet $A=\{1,2,\dots,p\}$ and a closed $\sigma$-invariant set $X\subset A^\mathbb{Z}$ (everything works just the same for one-sided shifts), let $\mathcal{L}=\mathcal{L}(X) \subset A^* = \bigcup_{n\geq 1} A^n$ be the collection of all finite words that appear in some sequence $x\in X$. Thus $X$ determines $\mathcal{L}$ and vice versa. Let $\mathcal{F} = A^* \setminus \mathcal{L}$ be the set of forbidden words. Now let $Y_n\subset A^\mathbb{Z}$ be the set of all sequences that do not contain any words in $\mathcal{F}$ of length $\leq n$. Then $Y_n$ is a shift of finite type and $X= \bigcap_{n\geq 1} Y_n$.

Furthermore, $Y_n$ is topologically transitive and hence intrinsically ergodic by virtue of being an SFT. To see this, choose any $v,w\in \mathcal{L}(Y_n)$ and write $v=v_1 v_2$, $w=w_1 w_2$ where $v_2$ and $w_1$ both have length exactly $n$. Then by transitivity of $X$ there exists a word $u$ such that $v_2 u w_1 \in \mathcal{L}(X)$, and by the definition of $Y_n$ we have $v u w\in \mathcal{L}(Y_n)$, which shows that $Y_n$ is transitive.

Thus the counterexample given in the answer to your earlier question works here as well. Actually, I'll point out that there are quite a broad class of such counterexamples, which can be constructed by looking at coded systems: these are shift spaces defined either in terms of a countable collection of generating words that are allowed to be freely concatenated, or equivalently in terms of a directed graph on countably many vertices with edges labeled from a finite alphabet. There are plenty of examples of coded systems that are transitive but not intrinsically ergodic (see this question or this answer, for example), and you can approximate coded systems from within by SFTs (or at least sofic shifts) in a very natural way: just truncate the collection of generators to a finite set, or truncate the graph to a finite subgraph.

In another direction, I believe there is a paper of Gurevich in which certain quantitative conditions are given on the rate of approximation from outside by intrinsically ergodic systems that turn out to be sufficient to guarantee intrinsic ergodicity of $X$. But I don't have the reference handy at the moment, and I'll have to wait until I'm in my office next week to dig up the paper and see if it's in fact relevant.

Edit: I found the Gurevich paper I was thinking of (actually 2 papers). References are as follows:

  1. B.M. Gurevic, "Uniqueness of the measure with maximal entropy for symbolic almost-Markov dynamic systems", Soviet Math. Dokl. 13 (1972), No. 3, 569-571.
  2. B.M. Gurevic, "Stationary random sequences of maximal entropy", Chapter 10 (pp. 327-380) of Multicomponent Random Systems, edited by R.L. Dobrushin and Ya.G. Sinai, Advances in Probability and Related Topics, Volume 6, Marcel Dekker Inc (1980).

As you see from the page count, (1) is quite short and just has the statement of the result, no proofs, while (2) is more comprehensive. Roughly speaking, the main result can be summarised as follows (the result in the paper is more precise because it doesn't assume that various limits exist).

Given a shift space $X$ on a finite alphabet, let $\mathcal{L}_n$ be the set of words of length $n$ that appear in some $x\in X$, and let $Y_n$ be the SFT defined by the condition that $x\in Y_n$ if and only if $x_k \cdots x_{k+n-1} \in \mathcal{L}_n$ for every $k$. Then $X = \bigcap_n Y_n$, and in particular, $h(Y_n) \to h(X)$, where $h$ is the topological entropy. Let $\rho_n = h(Y_n) - h(X)$ be the entropy gap; heuristically, $\rho_n$ is the amount of entropy that is destroyed by the restrictions in $X$ of length $>n$.

Furthermore, define $\alpha_n$ by $$ \alpha_n = \inf \{\tau \mid \forall u,v\in \mathcal{L}_n\ \exists w\in \mathcal{L}_\tau\ s.t. uvw\in \mathcal{L}\}. $$ That is, in the shift $X$, any two words of length $n$ can be glued together using a word of length $\alpha_n$. The shift $X$ has specification if and only if $\lim \alpha_n < \infty$, and in this case $X$ is intrinsically ergodic. (This is due to Bowen.)

Let $R_\alpha = \lim \frac 1n \log \alpha_n$ be the growth rate of $\alpha_n$, and let $R_\rho = -\lim \frac 1n \log \rho_n$ be the decay rate of $\rho_n$. Thus $R_\alpha$ is the rate at which the gluing time increases (and can be thought of as quantifying how badly $X$ fails to have specification), and $R_\rho$ is the rate at which the entropy gap decays (and can be thought of as quantifying how closely $X$ is approximated by the SFTs $Y_n$ in terms of entropy).

Theorem (Gurevich): If $h(X)>0$ and $R_\alpha < \frac{R_\rho} {16 h(X)}$, then $X$ is intrinsically ergodic.

Heuristically, "if the failure of specification is slow relative to the approximation by SFTs, then $X$ is intrinsically ergodic".

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Thanks Vaughn. I really will appreciate if you can send me the Gurevich reference when you have time. –  Rafael Alcaraz Jul 10 '12 at 16:18
I've added the Gurevich reference now - unfortunately I don't have the papers in an electronic version, but only in hard copy. In any case, I gave a rough statement of the result in my revised answer. –  Vaughn Climenhaga Jul 17 '12 at 20:39
Thanks Vaughn. I've looked the first reference. I'll try to find the second. –  Rafael Alcaraz Jul 31 '12 at 20:09
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