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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 …

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 …

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 …

Since you mention the related question of grouping together geodesics of lengths in the interval $[L,L+1]$, let me point out that this case is covered by the 1972 result of Bowen that I mentioned in t …

Burns and Donnay proved that every surface (including a sphere) admits a Riemannian metric that makes the geodesic flow ergodic with respect to Liouville measure, and hence topologically transitive (t …

This is true for every irrational $\theta$; the question can be rephrased in terms of Sturmian sequences. Given a sequence $z \in \{a,b\}^\mathbb{Z}$ and indices $i<j$, let $z_{[i,j]} \in \{a,b\}^{j- …

The most well-understood examples are the ones you mention: Axiom A diffeomorphisms and Markov maps of the interval, since these can be modeled by SFTs. Note that "Bernoulli" refers to a particular c …

The most comprehensive reference for this sort of thing seems to be Furstenberg, "Noncommuting random products", Trans. Amer. Math. Soc. 108 (1963), 377-428. I say this because I've seen it reference …