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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 …

There are two pretty simple proofs. Both rely on studying the action $T_* \colon \mathcal{M} \to \mathcal{M}$, where $\mathcal{M}$ is the space of Borel probability measures on $X$ and the action is …

For me the standard text is Peter Walters, "An Introduction to Ergodic Theory", Springer Graduate Texts in Mathematics.

The key words here are "large deviations"; large deviations theory addresses exactly this question. The answer depends quite a bit on the specific measure and system in question, but roughly speakin …

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 …

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 …

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 …

Every topologically transitive shift space, whether intrinsically ergodic or not, can be approximated from above by intrinsically ergodic systems. Indeed, given a finite alphabet $A=\{1,2,\dots,p\}$ …

There are lots of zero entropy invariant probability measures, many more than just the obvious ones supported on periodic orbits. As you suggest in the question, one can understand the general case b …

Even for Bernoulli shifts this property is not true, as the following example shows. Let $(X,\mu)$ be the $(\frac 12,\frac 12)$ Bernoulli shift on two symbols, 0 and 1. That is, $X = \{0,1\}^\mathbb{ …

Although it appears you've already settled matters with the information in Jon's answer, I'll offer a quick summary and elaboration. Let $(X,\mathcal{B},\mu)$ be a Lebesgue space (set + $\sigma$-alge …

There's a property called "entropy density of ergodic measures" (or variations on that terminology), which states that given an invariant measure μ, you can find a sequence of ergodic measures μn that …

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- …

No, there are many fully-supported non-ergodic measures. Just take a convex combination of a fully-supported measure and anything else. (Recall that an invariant measure is ergodic if and only if it …

I'll flesh out the consequences of Gerald's comment in a (CW-ed) answer. Lindenstrauss, Olsen, and Sternfeld showed in 1978 that if $S_1$ and $S_2$ are compact metrisable simplices such that the extr …