Ergodic theory of equivariant diffeomorphisms: Markov partitions and stable ergodicity (original) (raw)

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.

New criteria for ergodicity and non-uniform hyperbolicity

2009

In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of non-zero Lyapunov exponents in some contexts. In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C^1-dense among volume preserving partially hyperbolic diffeomorphisms with two-dimensional center bundle. This is motivated by a well known conjecture of C. Pugh and M. Shub.

Stable ergodicity of certain linear automorphisms of the torus

Annals of mathematics, 2005

We find a class of ergodic linear automorphisms of T N that are stably ergodic. This class includes all non-Anosov ergodic automorphisms when N = 4. As a corollary, we obtain the fact that all ergodic linear automorphism of T N are stably ergodic when N ≤ 5. of Jacob Palis. I am very grateful to him for many valuable commentaries and all his encouragement. I am also indebted to Mike Shub for patiently listening to the first draft of the proof and his helpful remarks. I wish to thank Enrique Pujals for several useful conversations and Raul Ures for showing me the dynamics of the pseudo-Anosov homeomorphisms of surfaces. Finally, I would like to thank the referees and Jana Rodriguez Hertz for helping me to improve the readability of this paper.

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.

DISCRETE AND CONTINUOUS UNIQUE ERGODICITY, STABLE ERGODICITY, AND THE MAUTNER PHENOMENON FOR DIFFEOMORPHISMS

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.

On the ergodicity of partially hyperbolic systems

Annals of Mathematics, 2010

Pugh and Shub have conjectured that essential accessibility implies ergodicity for a C 2 , partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satisfied in particular by all partially hyperbolic systems with 1-dimensional center bundle. We also obtain ergodicity results for C 1Cı partially hyperbolic systems.

Fine ergodic properties of partially hyperbolic dynamical systems

Let f : T 3 ! T 3 be a C 2 volume preserving partially hyperbolic diffeomorphism homotopic to a linear Anosov automorphism A : T 3 ! T 3. We prove that if f is Kolmogorov, then f is Bernoulli. We study the characteristics of atomic disintegration of the volume measure whenever it occurs. We prove that if the volume measure m has atomic disintegration on the center leaves then the disintegration has one atom per center leaf. We give a condition, depending only on the center Lyapunov exponent of the diffeomorphism, that guarantees atomic disintegration of the volume measure on center leaves. We construct an open family of diffeomorphisms satisfying this condition which generates the first examples of foliations which are both measurable and minimal. In this same construction we give the first examples of partially hyperbolic diffeomorphisms with zero center Lyapunov exponent and homotopic to a linear

On the Bernoulli property for certain partially hyperbolic diffeomorphisms

arXiv (Cornell University), 2016

We address the classical problem of equivalence between Kolmogorov and Bernoulli property of smooth dynamical systems. In a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to Anosov ("derived from Anosov") on 3-torus, we prove that Kolmogorov and Bernoulli properties are equivalent. In our approach, we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms. As an important consequence we obtain that there exists an almost everywhere conjugacy between any volume preserving derived from Anosov diffeomorphism of 3-torus and its linearization. Our results also hold in higher dimensional case when central bundle is one dimensional and stable and unstable foliations are quasi-isometric.