Quantitative recurrence for free semigroup actions

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'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.

A variational principle for free semigroup actions

“…Relative metric entropy of a semigroup action. In this section we will recall the notion of relative metric entropy proposed by Abramov and Rokhlin in [1], which was later adopted in [18] and [17] and whose connection with the topological entropy of the semigroup action was explored in [10].…”

“…Denote by M G the set of Borel probability measures on X invariant by g i , for all i ∈ {1, • • • , p}. In [10], it was shown that sup ν ∈ M G h (1) ν (S, P) ≤ h top (S) + (log p − h P (σ)) .…”

“…This inequality may, however, be strict, as exemplified in [10,Example 5.3]. In the next section we will introduce a more appropriate notion of metric entropy of a semigroup action.…”

“…Yet, we are looking for intrinsic dynamical and ergodic concepts, the less dependent on the skew product the better. In [20], the authors proposed a notion of measure theoretical entropy for free semigroup actions and probability measures invariant by all the generators of the semigroup, but only a partial variational principle was obtained there (as happened in [5] and [10]). The major obstruction to use this strategy, in order to extend the ergodic theory known for a single expanding map, is the fact that probability measures which are invariant by all the generators of a free semigroup action may fail to exist.…”

“…Following the idea of complexity presented in [8], a notion of topological entropy for a free semigroup action of Ruelle-expanding maps was introduced in [24], and its main properties have been described in [9] and [10] through a suitable transfer operator. In the former reference, the topological information has also been linked to ergodic measurements, both with respect to the relative metric entropy (cf.…”