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Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples  [PDF]
Christian Bonatti,Andrey Gogolev,Rafael Potrie
Mathematics , 2015,
Abstract: We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms $f$ on compact $3$-manifolds with fundamental groups of exponential growth such that $f^n$ is not homotopic to identity for all $n>0$. These provide counterexamples to a classification conjecture of Pujals.
Partial Hyperbolicity for Symplectic Diffeomorphisms  [PDF]
Ali Tahzibi,Vanderlei Horita
Mathematics , 2004,
Abstract: We prove that every robustly transitive and every stably ergodic symplectic diffeomorphism on a compact manifold admits a dominated splitting. In fact, these diffeomorphisms are partially hyperbolic.
Nonuniform Center Bunching and the Genericity of Ergodicity among $C^1$ Partially Hyperbolic Symplectomorphisms  [PDF]
Artur Avila,Jairo Bochi,Amie Wilkinson
Mathematics , 2008,
Abstract: We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns--Wilkinson and Avila--Santamaria--Viana. Combining this new technique with other constructions, we prove that $C^1$-generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms.
Diffeomorphisms with positive metric entropy  [PDF]
Artur Avila,Sylvain Crovisier,Amie Wilkinson
Mathematics , 2014,
Abstract: We obtain a dichotomy for $C^1$-generic, volume preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. hyperbolic and the splitting into stable and unstable spaces is dominated). We take this dichotomy as a starting point to prove a $C^1$ version of a conjecture by Pugh and Shub: among partially hyperbolic volume-preserving $C^r$ diffeomorphisms, $r>1$, the stably ergodic ones are $C^1$-dense. To establish these results, we develop new perturbation tools for the $C^1$ topology: "orbitwise" removal of vanishing Lyapunov exponents, linearization of horseshoes while preserving entropy, and creation of "superblenders" from hyperbolic sets with large entropy.
Topological Entropy and Partially Hyperbolic Diffeomorphisms  [PDF]
Yongxia Hua,Radu Saghin,Zhihong Xia
Mathematics , 2006,
Abstract: We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional, then the topological entropy is locally a constant; and if the center foliation is two dimensional, then the topological entropy is continuous on the set of all $C^\8$ diffeomorphisms. The proof uses a topological invariant we introduced; Yomdin's theorem on upper semi-continuity; Katok's theorem on lower semi-continuity for two dimensional systems and a refined Pesin-Ruelle inequality we proved for partially hyperbolic diffeomorphisms.
On m-minimal partially hyperbolic diffeomorphisms  [PDF]
Alexander Arbieto,Thiago Catalan,Felipe Nobili
Mathematics , 2015,
Abstract: We discuss about the denseness of the strong stable and unstable manifolds of partially hyperbolic diffeomorphisms. In this sense, we introduce a concept of m-minimality. More precisely, we say that a partially hyperbolic diffeomorphisms is m-minimal if m-almost every point in M has its strong stable and unstable manifolds dense in M. We show that this property has dynamics consequences: topological and ergodic. Also, we prove the abundance of m-minimal partially hyperbolic diffeomorphisms in the volume preserving and symplectic scenario.
Ergodic components of partially hyperbolic systems  [PDF]
Andy Hammerlindl
Mathematics , 2014,
Abstract: This paper gives a complete classification of the possible ergodic decompositions for certain open families of volume-preserving partially hyperbolic diffeomorphisms. These families include systems with compact center leaves and perturbations of Anosov flows under conditions on the dimensions of the invariant subbundles. The paper further shows that the non-open accessibility classes form a $C^1$ lamination and gives results about the accessibility classes of non-volume-preserving systems.
Robust ergodic properties in partially hyperbolic dynamics  [PDF]
Martin Andersson
Mathematics , 2008,
Abstract: We study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting. Earlier work of Bonatti, Viana about existence and finitude of physical measures is extended to the case of local diffeomorphisms. Moreover, we prove that such systems constitute a C^2-open set in which statistical stability is a dense property. In contrast, all mostly contracting systems are shown to be stable under small random perturbations.
New criteria for ergodicity and non-uniform hyperbolicity  [PDF]
F. Rodriguez Hertz,Jana Rodriguez Hertz,A. Tahzibi,R. Ures
Mathematics , 2009,
Abstract: 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.
$C^r$ density of stable ergodicity for a class of partially hyperbolic systems  [PDF]
Zhiyuan Zhang
Mathematics , 2015,
Abstract: We show that among a class of skew-product $C^{r}$ partially hyperbolic volume-preserving diffeomorphisms satisfying some pinching, bunching condition with certain type of dominated splitting in the centre subspace, a $C^{r}$ dense $C^{2}$ open subset contains ergodic diffeomorphisms for $r > 3$. As another application of our techniques, we partially generalised the result in \cite{DK} and obtain stable transitivity for action of random rotations on the sphere in arbitrary dimension.
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