Search Results

As indicated in the comments, measurability alone is not enough, and there are easy counterexamples. As for a condition between continuity and measurability that still does the trick, I'm not sure if …

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 …

The result on affinity of the entropy map holds for arbitrary finite convex combinations of invariant measures: if $\mu_1,\dots,\mu_n\in M(X,T)$ are any invariant measures and $a_1,\dots, a_n\in [0,1] …

If I remember my infinite ergodic theory correctly, any measure-preserving transformation $T$ of a $\sigma$-finite measure space $(\Omega,\mathcal{A},\mu)$ leads to a decomposition of $\Omega$ into a …

The language of the proof given in the book you refer to is a little different from the language I'm accustomed to, but I'll give what I believe is the exact same argument using a slightly different l …

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- …

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 …

I believe the Lemma you propose is true, via a relatively straightforward adaptation of the proof given for the additive case in Giles Atkinson, Recurrence of co-cycles and random walks, J. Lond. Math …

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 …

Poking through the DGS book mentioned in Ian's answer I came along a reference to a paper that turns out to be exactly what I wanted when I asked this question originally, so I'll post it here for the …

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{ …

Measures with the property you describe are called Bernoulli measures. There are many, many invariant measures that are not Bernoulli: one class of examples is given by the measures concentrated on …

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 …