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.
Ergodicity of non-autonomous discrete systems with non-uniform expansion
Discrete & Continuous Dynamical Systems - B, 2017
We study the ergodicity of non-autonomous discrete dynamical systems with nonuniform expansion. As an application we get that any uniformly expanding finitely generated semigroup action of C 1+α local diffeomorphisms of a compact manifold is ergodic with respect to the Lebesgue measure. Moreover, we will also prove that every exact non-uniform expandable finitely generated semigroup action of conformal C 1+α local diffeomorphisms of a compact manifold is Lebesgue ergodic.
In 1954, F. Mautner gave a simple representation theoretic argument that for compact surfaces of constant negative curvature, invariance of a function along the geodesic flow implies invariance along the horocycle flows (these are facts which imply ergodicity of the geodesic flow itself), [M]. Many generalizations of this Mautner phenomenon exist in representation theory, [St1]. Here, we establish a new generalization, Theorem 2.1, whose novelty is mostly its method of proof, namely the Anosov-Hopf ergodicity argument from dynamical systems. Using some structural properties of Lie groups, we also show that stable ergodicity is equivalent to the unique ergodicity of the strong stable manifold foliations in the context of affine diffeomorphisms.
A remark on conservative diffeomorphisms
Comptes Rendus Mathematique, 2006
We show that a stably ergodic diffeomorphism can be C 1 approximated by a diffeomorphism having stably non-zero Lyapunov exponents. Résumé. On montre qu'un difféomorphisme stablement ergodique peut être C 1 approché par un difféomorphisme ayant des exposants de Lyapunov stablement non-nuls. Two central notions in Dynamical Systems are ergodicity and hyperbolicity. In many works showing that certain systems are ergodic, some kind of hyperbolicity (e.g. uniform, non-uniform or partial) is a main ingredient in the proof. In this note the converse direction is investigated. Let M be a compact manifold of dimension d ≥ 2, and let µ be a volume measure in M . Take α > 0 and let Diff 1+α µ (M ) be the set of µ-preserving C 1+α diffeomorphisms, endowed with the C 1 topology. Let SE ⊂ Diff 1+α µ (M ) be the set of stably ergodic diffeomorphisms (i.e., the set of diffeomorphisms such that every sufficiently C 1 -close C 1+α conservative diffeomorphism is ergodic). Our result answers positively a question of [BuDP]:
3 Non-Removable Term Ergodic Action
2016
In this work, we introduce the concept of term ergodicity for action semigroups and construct semigroups on two dimensional manifolds which are C 1+α-robustly term ergodic. Moreover, we illustrate the term ergodicity by some exciting examples. At last, we study a problem in the context of circle packing which is concerned to term ergodic. 2000 Mathematics Subject Classification. 26A18, 37A99, 28A20. Key words and phrases. action semigroup, action group, minimality, ergodicity, robust property, circle packing. 1 A measure µ is said to be quasi-invariant if (gi) * µ is absolutely continuous with respect to µ for every element gi in G. lim r→0 Vol(A : B(x, r)) = Vol(A ∩ B(x, r)) Vol(B(x, r)) = 1, where B(x, r) is the geodesic ball of radius r centered about x. Denote by DP (A) the set of Lebesgue density points of a measurable set A. By Lebesgue density point theorem, for every measureable set A, Vol(A △ DP (A)) = 0. Now, consider a collection of diffeomorphisms G = {g 1 ,. .. , g s } on M. Write G −1 = {g −1 1 ,. .. , g −1 s }. The action semigroup < G > + generated by G is given by < G > + = {h : M → M : h = g in • • • • • g i 1 , i j ∈ {1,. .. , s}} ∪ {id : M → M }.
Mathematics of Complexity and Dynamical Systems, 2012
Contents Glossary 1 Definition and Importance of the Subject 2 1. Introduction 2 2. The volume class 3 3. The fundamental questions 4 4. Lebesgue measure and local properties of volume 4 5. Ergodicity of the basic examples 6 6. Hyperbolic systems 8 7. Beyond uniform hyperbolicity 15 8. The presence of critical points and other singularities 19 9. Future directions 22 References 22 Glossary Conservative, Dissipative: Conservative dynamical systems (on a compact phase space) are those that preserve a finite measure equivalent to volume. Hamiltonian dynamical systems are important examples of conservative systems. Systems that are not conservative are called dissipative. Finding physically meaningful invariant measures for dissipative maps is a central object of study in smooth ergodic theory.
On the ergodic theory of free group actions by real-analytic circle diffeomorphisms
Inventiones mathematicae, 2017
We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due toÉ. Ghys, G. Hector and D. Sullivan.
Stably ergodic diffeomorphisms which are not partially hyperbolic
Israel Journal of Mathematics, 2004
We show stable ergodicity of a class of conservative diffeomorphisms of T n which do not have any hyperbolic invariant subbundle. Moreover, the uniqueness of SRB (Sinai-Ruelle-Bowen) measure for non-conservative C 1 perturbations of such diffeomorphisms is verified. This class strictly contains non-partially hyperbolic robustly transitive diffeomorphisms constructed by Bonatti-Viana [BV00] and so we answer the question posed there on the stable ergodicity of such systems.