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 $i=1,2$?