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I'll offer a few partial answers, which may eventually lead to a complete answer. As observed in Saussol's paper (Theorem 3, Kac's lemma), if you have an ergodic invariant measure $\mu$ then the mean …

Regarding your original question about Birkhoff averages, the story is the following: Suppose $X$ is a compact metric space, $T\colon X\to X$ is continuous, and $f\colon X\to \mathbb{R}$ is continuous …

The following doesn't necessarily answer your question about how to intuitively interpret entropy and dimension, but it does address the relationship between them, at least in the symbolic setting. In …

The limit exists for the first two examples that come to mind, namely topological entropy on the full shift and on certain simple Markov shifts. If $X \subset \Sigma_d^+ = \{1,2, \dots, d\}^{\mathbb{N …

In addition to Igor's answer, there's also: D. Ornstein and B. Weiss, "The Shannon–McMillan–Breiman theorem for a class of amenable groups", Israel J. Math. 44 (1983), 53–60. Zbl 0516.28020

Yes. This is a consequence of the following simple exercise about exponential growth: if $a_n\geq 0$ is any sequence such that $P = \lim \frac 1n \log a_n$ exists, then $\frac 1n \log \sum_{k=1}^n a_k …

Edit: I've updated this answer to reflect the helpful comments made by Andres Koropecki and Ian Morris. As the other answers mentioned, the first crucial distinction you must make is that some proper …

Suppose $X$ is the unit circle and $\varphi$ is the doubling map (multiplicatively, $X = \{ z\in \mathbb{C} : |z| = 1\}$ and $\varphi(z) = z^2$, or additively, $X = \mathbb{R}/\mathbb{Z}$ and $\varphi …

The measure $\mu$ does not necessarily have the Gibbs property. In fact, it has the Gibbs property if and only if $f$ has the Bowen property: $\sup_n \sup \{ |S_n f(x) - S_n f(y)| : x_1 \dots x_n = y_ …

Yes. The simplest construction is to let $f$ be the figure-eight system so that $\delta_p$ is a physical non-SRB measure (where $p$ is the saddle point) and let $g$ be an Anosov diffeomorphism with SR …

The most intuitive explanation I know is the following: suppose that you have a certain amount of mass (I usually picture a pile of sand) that is distributed over $\Sigma_A^+$ according to the density …

I realize this doesn't directly answer the "reference request" part of the question, but I believe that if you require $V$ to be full (Lebesgue) measure in a neighborhood of the support of $\mu$, then …

The SRB measure is always isomorphic to a Bernoulli scheme (up to a period) and hence has positive entropy. Regarding continuity properties of entropy, we have upper semicontinuity whenever the map …

In the setting you describe, for each $\alpha \in (0,1)$ the $(1-\alpha,\alpha)$-Bernoulli measure is the unique measure achieving the maximum. The function $\alpha \mapsto \eta(\alpha)$ is the Legen …

So far as I know the best result you can hope for in full generality is the Shannon-McMillan-Breiman Theorem that you quote: If $(X,\sigma)$ is a shift space and $\mu$ is an ergodic shift-invariant me …