On ergodicity of Markovian mostly expanding semi-group actions

Abstract

In this work, we address ergodicity of smooth actions of finitely generated semigroups on an m-dimensional closed manifold M . We provide sufficient conditions for such an action to be ergodic with respect to the Lebesgue measure. Our results improve the main result in [8], where the ergodicity for one dimensional fiber was proved. We will introduce Markov partition for finitely generated semi-group actions and then we establish ergodicity for a large class of finitely generated semi-groups of C 1+α -diffeomorphisms that admit a Markov partition.Moreover, we present some transitivity criteria for semi-group actions and provide a weaker form of dynamical irreducibility that suffices to ergodicity in our setting.

Ergodicity of non-autonomous discrete systems with non-uniform expansion

“…In fact, we will show that there are no nonuniformly expanding finitely generated semigroup actions of diffeomorphisms. Because of this, in [10] the authors introduced a weak form of non-uniform expansion. Namely, they ask the existence of a constant a > 0 such that for m-almost every x ∈ M there is ω ∈ Ω such that lim sup…”

“…Proposition 2.13. Assume that m is f 1,∞ -conformal as above and there exists k ∈ N, 0 < α ≤ 1, ǫ > 0 such that the functions (ψ n ) n satisfy the locally Hölder condition (10). Then any (δ, λ)-preball of a point (k, x) ∈ M with 0 < δ ≤ ǫ, 0 < λ < 1 has bounded distortion, i.e., satisfies (5) with distortion constant K = K(δ, λ, C k ) uniform on x and on the order of the preball.…”