# Periodic points and measures for a class of skew products

## TL;DR

This paper studies a class of skew product dynamical systems, showing that generically they have unique measures of maximal entropy, hyperbolic periodic points, and rich invariant sets, with some systems also preserving a natural physical measure.

## Contribution

It establishes generic properties of measures and periodic points for a class of partially hyperbolic skew products, including uniqueness of maximal entropy measures and symbolic extensions.

## Key findings

- Existence of a unique measure of maximal entropy for generic systems.
- Presence of hyperbolic periodic points and invariant sets with high entropy.
- Systems with a residual subset have equal topological and periodic entropies.

## Abstract

We consider the open set constructed by M. Shub in [42] of partially hyperbolic skew products on the space $\mathbb{T}^2\times \mathbb{T}^2$ whose non-wandering set is not stable. We show that there exists an open set $\mathcal{U}$ of such diffeomorphisms such that if $F_S\in \mathcal{U}$ then its measure of maximal entropy is unique, hyperbolic and, generically, describes the distribution of periodic points. Moreover, the non-wandering set of such an $F_S\in \mathcal{U}$ contains closed invariant subsets carrying entropy arbitrarily close to the topological entropy of $F_S$ and within which the dynamics is conjugate to a subshift of finite type. Under an additional assumption on the base dynamics, we verify that $F_S$ preserves a unique SRB measure, which is physical, whose basin has full Lebesgue measure and coincides with the measure of maximal entropy. We also prove that there exists a residual subset $\mathcal{R}$ of $\mathcal{U}$ such that if $F_S\in \mathcal{R}$ then the topological and periodic entropies of $F_S$ are equal, $F_S$ is asymptotic per-expansive, has a sub-exponential growth rate of the periodic orbits and admits a principal strongly faithful symbolic extension with embedding.