Rational functions with identical measure of maximal entropy

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.

Maximal Entropy Measure for Rational Maps and a Random Iteration Algorithm for Julia Sets

We prove that for any rational map of degree d, the unique invariant measure of entropy log d can be obtained as the weak limit of atomic probability measures obtained by taking a random backward orbit along an arbitrary nonexceptional point. This proves that the computer algorithm suggested by Barnsley in some specific cases works for all rational maps and is the algorithm used to produce Juli...

Some Transformations Having a Unique Measure with Maximal Entropy

1. Introduction Let X be a compact metric space and T: X-> X a homeomorphism of X onto X. Let M(T) denote the collection of all T-invariant Borel probability measures on X. By Krylov and Bogolioubov's work we know M(T) is non-empty (see [10]). M{T) is a convex set and closed in the weak topology. For /x e M(T), h(T, p) will denote the measure-theoretic entropy of T with respect to p. Ifh top (T...

C Surface Diffeomorphisms with No Maximal Entropy Measure

For any 1 ≤ r <∞, we build on the disk and therefore on any manifold, a C-diffeomorphism with no measure of maximal entropy. Résumé. Pour tout 1 ≤ r < ∞, nous construisons, sur le disque et donc sur toute variété, un difféomorphisme de classe C sans mesure d’entropie maximale.

A Measure of Maximal Entropy for Nonuniformly Hyperbolic Hilbert Geometries