On the generic behavior of the metric entropy, and related quantities, of uniformly continuous maps over Polish metric spaces

TL;DR

This paper investigates the generic properties of invariant measures with zero metric entropy and related quantities for uniformly continuous maps on Polish metric spaces, confirming conjectures and extending results to expansive and Lipschitz maps.

Contribution

It proves that the set of zero-entropy invariant measures is generic under certain conditions, confirming Sigmund's conjecture and extending results to expansive and Lipschitz maps.

Findings

The set of zero-entropy invariant measures is a $G_\delta$ set when periodic measures are dense.

Typically, lower correlation entropy for $q\in(0,1)$ is zero for expansive or Lipschitz maps with dense periodic measures.

The set of invariant measures with zero packing dimension and recurrence indicators is residual for expanding maps with dense periodic measures.

Abstract

In this work, we show that if $f$ is a uniformly continuous map defined over a Polish metric space, then the set of $f$-invariant measures with zero metric entropy is a $G_\delta$ set (in the weak topology). In particular, this set is generic if the set of $f$-periodic measures is dense in the set of $f$-invariant measures. This settles a conjecture posed by Sigmund (Sigmund, K. On dynamical systems with the specification property. Trans. Amer. Math. Soc. 190 (1974), 285-299) which states that the metric entropy of an invariant measure of a topological dynamical system that satisfies the periodic specification property is typically zero. We also show that if $X$ is compact and if $f$ is an expansive or a Lipschitz map with a dense set of periodic measures, typically the lower correlation entropy for $q\in(0,1)$ is equal to zero. Moreover, we show that if $X$ is a compact metric space and if $f$ is an expanding map with a dense set of periodic measures, then the set of invariant measures with packing dimension, upper rate of recurrence and upper quantitative waiting time indicator equal to zero is residual. Finally, we present an alternative proof of the fact that the set of expansive measures is a $G_{\delta\sigma}$ set in the set of probability measures $\M(X)$, if $X$ is a Polish metric space and if $f$ is uniformly continuous; this result was originally proved by Lee, Morales and Shin (Lee, K., Morales, C. A., and Shin, B. On the set of expansive measures. Commun. Contemp. Math. 20, 7 (2018), 1750086, 10) for compact metric spaces.