Semigroup Actions of Expanding Maps

Abstract

We consider semigroups of Ruelle-expanding maps, parameterized by random walks on the free semigroup, with the aim of examining their complexity and exploring the relation between intrinsic properties of the semigroup action and the thermodynamic formalism of the associated skew-product. In particular, we clarify the connection between the topological entropy of the semigroup action and the growth rate of the periodic points, establish the main properties of the dynamical zeta function of the semigroup action and prove the existence of stationary probability measures.

On the measure-theoretic entropy and topological pressure of free semigroup actions

“…Biś [3] and Bufetov [8] introduced the notion of the topological entropy of free semigroup actions. Related studies include [6,11,23,24,27,29,31,32].…”

Quantitative recurrence for free semigroup actions

“…The action of semigroups of dynamics has a strong connection with skew products which has been scanned in order to obtain properties of semigroup actions by means of fibred and annealed quantities associated to the skew product dynamics (see e.g. [10]). We recall that, if X is a compact metric space and one considers a finite set of continuous maps g i : X → X, i ∈ P = {1, 2, .…”

“…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]).…”

“…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]. The main strategy to deal with such a system has been the codification of the random orbits by a true dynamics, namely the skew product based on a full shift with finitely many symbols.…”

A variational principle for free semigroup actions

“…More precisely, after fixing a finite set {g 1 , g 2 , • • • , g p } of endomorphisms of X and taking the unilateral shift σ : Σ + p → Σ + p defined on the space of sequences with values in {1, 2, • • • , p}, endowed with a Borel σ−invariant probability measure P, we associate to each ω = ω 1 ω 2 • • • ∈ Σ + p the sequence of compositions (g ω 1 g ω 2 • • • g ωn ) n ∈ N . Our aim is to carry on the analysis, started in [9,24], of the ergodic and statistical properties of these random compositions, and to set up a thermodynamic formalism. In this context common invariant measures seldom exist.…”