Uniformly Positive Entropy of Induced Transformations

TL;DR

This paper establishes an equivalence between uniformly positive entropy of a topological dynamical system and its induced system on the space of probability measures, extending a known result for topological entropy.

Contribution

It proves that uniformly positive entropy is preserved under the induced measure system, generalizing Glasner and Weiss's result for topological entropy.

Findings

Uniformly positive entropy of (X,T) iff of ((X), T)

Uses local entropy theory for proof

Extends Glasner and Weiss's result

Abstract

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space $X$ and a continuous surjective map $T : X \to X$. By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $(\cM(X),\wt{T})$ on the space of Borel probability measures endowed with the weak$^*$ topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.