Quenched and annealed equilibrium states for random Ruelle expanding maps and applications

Abstract

We find generalized conformal measures and equilibrium states for random dynamics generated by Ruelle expanding maps, under which the dynamics exhibits exponential decay of correlations. This extends results by Baladi [Correlation spectrum of quenched and annealed equilibrium states for random expanding maps. Comm. Math. Phys.186 (1997), 671–700] and Carvalho et al [Semigroup actions of expanding maps. J. Stat. Phys.116(1) (2017), 114–136], where the randomness is driven by an independent and identically distributed process and the phase space is assumed to be compact. We give applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and introduce a boundary of equilibria for not necessarily free semigroup actions.

Entropy inequalities for semigroup actions

“…[20]) or if η a is replaced by another Markov probability measure (cf. [41]). Moreover, it was proved in [16,Section 8] that h top (S, η a ) = P (a)…”