Generic points and local entropies - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:22:28Z http://mathoverflow.net/feeds/question/56215 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56215/generic-points-and-local-entropies Generic points and local entropies Vaughn Climenhaga 2011-02-21T21:07:29Z 2011-11-30T05:22:12Z <p>Let $X=\{1,\dots,p\}^\mathbb{N}$ be the space of sequences on a finite alphabet with a metric inducing the product topology, and let $\sigma\colon X\to X$ be the shift map. Let $\mu$ be a $\sigma$-invariant Borel probability measure on $X$.</p> <p>A point $x\in X$ is <em>generic</em> for $\mu$ if $\frac 1n S_n\phi(x) \to \int \phi\,d\mu$ for every $\phi\in C(X)$, where $S_n \phi(x) = \phi(x) + \phi(\sigma x) + \cdots + \phi(\sigma^{n-1}x)$. Denote by $G_\mu$ the set of $\mu$-generic points.</p> <p><strong>Fact #1.</strong> If $\mu$ is ergodic, then Birkhoff's ergodic theorem implies that $\mu(G_\mu)=1$.</p> <hr> <p>The <em>local entropy</em> of a point $x\in X$ is $h_\mu(x) = \lim_{n\to\infty} -\frac 1n \log \mu([x_1\dots x_n])$, where $[x_1 \dots x_n] = \{ y\in X \mid y_i = x_i \,\forall 1\leq i\leq n\}$, provided the limit exists. Denote by $Z_\mu$ the set of points $x$ for which $h_\mu(x)$ exists and is equal to the measure-theoretic entropy $h_\mu(\sigma)$.</p> <p><strong>Fact #2.</strong> If $\mu$ is ergodic, then Shannon-McMillan-Breiman implies that $\mu(Z_\mu)=1$.</p> <hr> <p>The measure $\mu$ is a Gibbs measure if there exists a function $\phi\in C(X)$ and constants $K,P>0$ such that $$K^{-1} \leq \frac{\mu([x_1\dots x_n])}{e^{-nP + S_n \phi(x)}} \leq K$$ for every $x\in X$ and $n\in \mathbb{N}$.</p> <p><strong>Fact #3.</strong> If $\mu$ is a Gibbs measure, then $G_\mu \subset Z_\mu$. That is, the local entropy of a point $x$ with respect to $\mu$ is "what it should be" provided the Birkhoff averages of continuous functions along the orbit of $x$ are "what they should be".</p> <p>(Actually, even more is true: for a Gibbs measure the local entropy $h_\mu(x)$ of any point $x$ is completely determined by the Birkhoff averages $\frac 1n S_n \phi(x)$ of a single function.)</p> <hr> <p><strong>Question.</strong> What is the broadest class of measures for which the inclusion $G_\mu \subset Z_\mu$ holds -- that is, for which genericity for Birkhoff averages of continuous functions implies genericity for local entropies? Does this hold for <em>all</em> ergodic measures? If it does not, is there a natural class of measures beyond the Gibbs measures (and various notions of weak Gibbs measures) for which it does hold?</p> <p><strong>Related question.</strong> Gibbs measures (and weak Gibbs measures) have the property that there exists a function $\phi\in C(X)$ such that $h_\mu(x)$ is completely determined by $\frac 1n S_n \phi(x)$ for <em>every</em> $x\in X$. (Not just for a full measure set of $x$ -- this is true for all ergodic measures.) Is there an example of a measure $\mu$ such that there is no single function $\phi\in C(X)$ whose Birkhoff averages determine $h_\mu(x)$, but there exist <em>two</em> function $\phi_1, \phi_2\in C(X)$ such that $h_\mu(x)$ is completely determined by $\frac 1n S_n \phi_i(x)$ for $i=1,2$?</p> http://mathoverflow.net/questions/56215/generic-points-and-local-entropies/56274#56274 Answer by Ian Morris for Generic points and local entropies Ian Morris 2011-02-22T12:32:13Z 2011-02-22T12:32:13Z <p>My feeling is that there exists an ergodic measure $\mu$ for which $G_\mu \setminus Z_\mu$ is nonempty. It is sufficient to find a uniquely ergodic subsystem which admits exceptional points for the Shannon-McMillan-Breiman theorem. I think that one can be constructed symbolically without too much difficulty by the following method.</p> <p>Pick a real number $h$ lying strictly between 0 and $\log 2$, and consider a sequence $x$ in the 2-shift with the following properties:</p> <p>1) For every $n \geq 1$, the sequence contains precisely $e^{nh + o(n)}$ distinct words of length $n$. (For reasons of subadditivity the $o(n)$ term is necessarily positive).</p> <p>2) Every word which occurs in $x$ occurs with a well-defined frequency which is not equal to 0 or 1.</p> <p>The orbit closure $X$ of such a sequence is then a uniquely ergodic subsystem of the shift with topological entropy equal to $h$. An explicit procedure for constructing such a sequence was given by Grillenberger in the 1970s (in my opinion it's not particularly hard). In particular, $X$ supports a unique invariant measure $\mu$ and $G_\mu$ includes the whole of $X$. Now, suppose that the word $x$ also satisfies the property:</p> <p>3) There exists a nested sequence of subwords of $x$ such that the frequency of each of these words is less than $e^{-n(h+\varepsilon)}$ for some $\varepsilon>0$.</p> <p>This implies that there is a nested sequence of cylinder sets in $X$, containing some point, such that the measures of these cylinder sets decrease at a rate faster than the "standard" local entropy $h$, and hence the point in the intersection of the cylinders belongs to $G_\mu$ but not to $Z_\mu$.</p> <p>I think that there shouldn't be any problem in reconciling all three of these criteria with one another, but I will admit that I haven't attempted to write a proof of that. I think it sounds reasonable that for a larger class of measures than Gibbs measures we should have $G_\mu \subseteq Z_\mu$, but I don't have much to contribute to that end of the question...</p>