3 Ergodicity of Expanding Minimal Actions (original) (raw)

Expanding actions: minimality and ergodicity

Stochastics and Dynamics, 2017

We prove that every expanding minimal semigroup action of [Formula: see text] diffeomorphisms of a compact manifold (resp. [Formula: see text] conformal) is robustly minimal (resp. ergodic with respect to the Lebesgue emeasure). We also show how, locally, a blending region yields the robustness of the minimality and implies ergodicity.

Ergodicity of expanding minimal actions

We prove that every expanding minimal group action of C1+alphaC^{1+\alpha}C1+alpha-diffeomorphisms of a compact manifold is robustly ergodic with respect to the Lebesgue measure. We also demonstrate how, locally, a blending region yields the robustness of both, minimality and ergodicity. Similar results are also obtained for semigroup actions.

On ergodicity of mostly expanding semi-group actions

In this work, we address ergodicity of smooth actions of finitely generated semi-groups 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+alpha}-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.

On the question of ergodicity for minimal group actions on the circle

2008

This work is devoted to the study of minimal, smooth actions of finitely generated groups on the circle. We provide a sufficient condition for such an action to be ergodic (with respect to the Lebesgue measure), and we illustrate this condition by studying two relevant examples. Under an analogous hypothesis, we also deal with the problem of the zero Lebesgue measure for exceptional minimal sets. This hypothesis leads to many other interesting conclusions, mainly concerning the stationary and conformal measures. Moreover, several questions are left open. The methods work as well for codimension-one foliations, though the results for this case are not explicitly stated.

Stable local dynamics: expansion, quasi-conformality and ergodicity

2021

In this paper, we study the stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. The existence of such actions is known only on one-dimensional manifolds. The aim of this paper is to overcome this restriction and to give a method for constructing higher-dimensional examples. In particular, we show that every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class C. We also prove the stable ergodicity of certain algebraic actions including the natural action of a generic pair of matrices near the identity on a sphere of arbitrary dimension. These are consequences of a new local and stable mechanism/phenomenon which we call quasi-conformal blender. This tool stably provides quasi-conformal orbits and yields stable local ergodicity. The quasi-conformal blender is developed in the context of pseudo-semigroup actions of locally defined smooth diffeomorphisms which allows for applications in several different settings.

A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms

arXiv preprint arXiv:0710.2353, 2007

In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the C 1 topology, to a conjecture of Pugh-Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle.