Invariance principle and rigidity of high entropy measures

Metric Invariance Entropy and Conditionally Invariant Measures

A notion of metric invariance entropy is constructed with respect to a conditionally invariant measure for control systems in discrete time. It is shown that the metric invariance entropy is invariant under conjugacies, the power rule holds, and the (topological) invariance entropy provides an upper bound. November 20, 2014 Key words. invariance entropy, conditionally invariant measures, quasi-...

Measures of maximal entropy

We extend the results of Walters on the uniqueness of invariant measures with maximal entropy on compact groups to an arbitrary locally compact group. We show that the maximal entropy is attained at the left Haar measure and the measure of maximal entropy is unique.

Maximum Entropy Principle with General Deviation Measures

An approach to the Shannon and Rényi entropy maximization problems with constraints on the mean and law invariant deviation measure for a random variable has been developed. The approach is based on the representation of law invariant deviation measures through corresponding convex compact sets of nonnegative concave functions. A solution to the problem has been shown to have an alpha-concave d...

Rigidity of Measures – the High Entropy Case, and Non-commuting Foliations

We consider invariant measures for partially hyperbolic, semisimple, higher rank actions on homogeneous spaces defined by products of real and p-adic Lie groups. In this paper we generalize our earlier work to establish measure rigidity in the high entropy case in that setting. We avoid any additional ergodicity-type assumptions but rely on, and extend the theory of conditional measures.

On Modular Invariance and Rigidity Theorems