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

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 (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.