Quantitative recurrence for free semigroup actions**MC has been financially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER) under the partnership agreement PT2020. FR and PV were partially supported by BREUDS. PV has also benefited from a fellowship awarded by CNPq-Brazil and is grateful to the Faculty of Sciences of the

Abstract

We consider finitely generated free semigroup actions on a compact metric space and obtain quantitative information on Poincaré 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 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.