Maximizing entropy measures for random dynamical systems

Abstract

We prove the existence of relative maximal entropy measures for certain random dynamical systems of the type [Formula: see text], where [Formula: see text] is an invertibe map preserving an ergodic measure [Formula: see text] and [Formula: see text] is a local diffeomorphism of a compact Riemannian manifold exhibiting some non-uniform expansion. As a consequence of our proofs, we obtain an integral formula for the relative topological entropy as the integral of the logarithm of the topological degree of [Formula: see text] with respect to [Formula: see text]. When [Formula: see text] is topologically exact and the supremum of the topological degree of [Formula: see text] is finite, the maximizing measure is unique and positive on open sets.

Uniqueness and Stability of Equilibrium States for Random Non-uniformly Expanding Maps

“…The main tool in the proof of Proposition 6.3 is the existence of a generating partition for the equilibrium state. In the context of random dynamical systems generated by non-uniformly expanding maps, the existence of generating partitions for ergodic measures with Lyapunov exponents bounded away from zero was proved by Bilbao and Oliveira in [8]. Therefore in this setting we can also apply our Proposition 6.3 to obtain uniqueness of equilibrium states.…”

“…Let P α be the Bernoulli measure on the sequence space X = {0, 1} Z such that P α ([1]) = α. In [8] was proved the existence of α ∈ (0, 1) close to 1 such that lim C k (w) dP α (w) > 0 which means that the hypothesis of Corollary 6.1 was verified. Thus, for potentials φ ∈ L 1 P (X, C α (M )) satisfying (IV) such that sup φ w − inf φ w < ε log deg(f w )dP α we conclude uniqueness of equilibrium states.…”

“…Next we present an application of our Corollary 6.1. This example appears in [8] in the context of maximizing entropy measures. Here we prove uniqueness of equilibrium states for potentials with small variation.…”

“…The existence of equilibrium states with positive Lyapunov exponents was proved by Arbieto, Matheus and Oliveira [5] for certain non-uniformly expanding maps and continuous potentials with low variation. In [8] Bilbao and Oliveira obtained uniqueness of maximizing entropy measures in this context. Recently, Stadlbauer, Suzuki and Varandas [25] developed the thermodynamical formalism for a wide class of random maps with non-uniform expansion and diferentiable potentials at high temperature.…”

Thermodynamic formalism for random non-uniformly expanding maps

“…In particular, measures of maximal entropy do exist and, under a transitivity assumption, it is unique (cf. [8]). Much more recently, Atnip et al [5] constructed equilibrium states for random covering interval maps (a condition defined in terms of the potential and transfer operators acting on BV spaces), covering important examples as random beta-maps and random Liverani-Saussol-Vaienti maps.…”