Quantitative recurrence for free semigroup actions

TL;DR

This paper studies recurrence properties and entropy in free semigroup actions on compact spaces, providing quantitative results and linking recurrence rates to expansion and entropy, with implications for ergodic optimization.

Contribution

It introduces new quantitative recurrence results for free semigroup actions, connecting recurrence rates to generator expansion and topological entropy, and establishes a variational principle.

Findings

Quantitative recurrence estimates for free semigroup actions.

Relation between recurrence rates, generator expansion, and entropy.

Partial variational principle and ergodic optimization results.

Abstract

We consider finitely generated free semigroup actions on a compact metric space and obtain quantitative information on Poincar\'e recurrence, average first return time and hitting frequency for the random orbits induced by the semigroup action. Besides, we relate the recurrence to balls with the rates of expansion of the semigroup's generators and the topological entropy of the semigroup action. Finally, we establish a partial variational principle and prove an ergodic optimization for this kind of dynamical action.