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

Silas L. Carvalho and Alexander Condori

Mathematische Nachrichten, 2023, vol. 296, issue 3, 980-995

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 Gδ$G_\delta$ 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 (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∈(0,1)$q\in (0,1)$ 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.

Date: 2023
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