Generic points and local entropies

Question

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 $X$.

A point $x\in X$ is generic 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.

Fact #1. If $\mu$ is ergodic, then Birkhoff's ergodic theorem implies that $\mu(G_\mu)=1$.

---

The local entropy 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 $h_\mu(\sigma)$.

Fact #2. If $\mu$ is ergodic, then Shannon-McMillan-Breiman implies that $\mu(Z_\mu)=1$.

---

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 $n\in \mathbb{N}$.

Fact #3. 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".

(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.)

---

Question. 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 all 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?

Related question. 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 every $x\in X$. (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 two 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 $i=1,2$?

Answer 1

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.

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:

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).

2) Every word which occurs in $x$ occurs with a well-defined frequency which is not equal to 0 or 1.

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:

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 $\varepsilon>0$.

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 $Z_\mu$.

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...