Specification and thermodynamical properties of semigroup actions

Abstract

Abstract. In the present paper we study the thermodynamical properties of finitely generated continuous subgroup actions. We propose a notion of topological entropy and pressure functions that does not depend on the growth rate of the semigroup and introduce strong and orbital specification properties, under which, the semigroup actions have positive topological entropy and all points are entropy points. Moreover, we study the convergence and Lipschitz regularity of the pressure function and obtain relations between topological entropy and exponential growth rate of periodic points in the context of semigroups of expanding maps, obtaining a partial extension of the results obtained by Ruelle for Z d -actions [33] . The specification properties for semigroup actions and the corresponding one for its generators and the action of push-forward maps is also discussed.

Quantitative recurrence for free semigroup actions

“…Meanwhile, recall that P (q) top (F G , 0, P) stands for the quenched topological pressure of the skew product F G with respect to the random walk P (see [5]), h top (S) is the topological entropy of the free semigroup action S (cf. definition in [21]) and h top (S, P) is the relative topological entropy of the free semigroup action with respect to the random walk P (see [10]).…”

“…We will start this section recalling the notion of orbital specification property introduced in [21] and a few facts about recurrence by the skew product associated to a free semigroup action. The reader acquainted with this preliminary information may omit the next two subsections.…”

“…We proceed covering the dynamical ball B ω δ (x, n) ⊂ X by a collectionβ n 0 (ω) of partition elements in β n 0 (ω). If δ > 0 is chosen small enough, then (21) implies that the piece of orbit (f j ω (x)) n j=0 enters the δ-neighborhood of ∂β in at most γ n iterates. The argument used in [25, Lemma 3.2] implies that, for any α > 0, there exist γ > 0 and δ > 0 (given by (21)) so that B ω δ (x, n) ⊂ X is covered by a collectionβ n 0 (ω) of at most e αn partition elements of β n 0 (ω), for every x ∈ E ω ξ .…”

“…Ultimately, we are concerned with the description of the fastest return time when considering all the semigroup elements instead of a single dynamical system. In a recent work [21], it has been introduced a notion of topological entropy and pressure for finitely generated continuous free semigroup actions on a compact metric space. Later, in [10], it has been shown that a free semigroup action of either C 1 expanding maps, or, more generally, Ruelle-expanding transformations, has a unique measure of maximal entropy which is linked to annealed equilibrium states for random dynamical systems [5].…”

Weak mixing in switched systems

“…When the switchings are arbitrary, one can take the switched system (1.1) as a free semigroup action G generated by G, i.e., G = n∈N G n , G n = {f ωn · · · f ω 1 | ω i ∈ I, i = 1, · · · , n}. There are many works for dynamical systems under semigroup actions: topological entropy [4,6,30,31]; transitivity, mixing, and chaos [7,28,29,33,27,12]; sensitivity [14,24]; shadowing property [2,12]; specification property [25,12]; we refer the reader to the references therein for other related investigations.…”