Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center

Science China Mathematics - In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of...     

Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

We show that partially hyperbolic diffeomorphisms of \(d\) -dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a global stability result, i.e. every partially hyperbolic diffeomorphism as above is leaf-conjugate to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.     

Transitive partially hyperbolic diffeomorphisms on 3-manifolds

The known examples of transitive partially hyperbolic diffeomorphisms on 3-manifolds belong to 3 basic classes: perturbations of skew products over an Anosov map of T², perturbations of the time one map of a transitive Anosov flow, and certain derived from Anosov diffeomorphisms of the torus T³. In this work we characterize the two first types by a local hypothesis associated to one closed periodic curve.     

Topological and ergodic aspects of partially hyperbolic diffeomorphisms and nonhyperbolic step skew products

We review some ergodic and topological aspects of robustly transitive partially hyperbolic diffeomorphisms with one-dimensional center direction. We also discuss step skew-product maps whose fiber maps are defined on the circle which model such dynamics. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents as well as intermingled horseshoes having different types of hyperbolicity. We discuss some recent advances concerning the topology of the space of invariant measures and properties of the spectrum of Lyapunov exponents.     

Codimension one generic homoclinic classes with interior

We study C ¹-generic diffeomorphisms with a homoclinic class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive.     

New partially hyperbolic dynamical systems I

We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms. Laying aside many surgery constructions of 3-dimensional Anosov flows, these are the first new examples of manifolds which admit partially hyperbolic diffeomorphisms in the past forty years.     

Nonuniform hyperbolicity for C1-generic diffeomorphisms

We study the ergodic theory of non-conservative C ¹-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C ¹-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C ¹-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.     

Stably ergodic diffeomorphisms which are not partially hyperbolic

We show stable ergodicity of a class of conservative diffeomorphisms ofT ⁿ which do not have any hyperbolic invariant subbundle. Moreover, the uniqueness of SRB (Sinai-Ruelle-Bowen) measure for non-conservativeC ¹ perturbations of such diffeomorphisms is verified. This class strictly contains non-partially hyperbolic robustly transitive diffeomorphisms constructed by Bonatti-Viana [4] and so we answer the question posed there on the stable ergodicity of such systems.     

SRB measures for partially hyperbolic systems whose central direction is mostly contracting

We consider partially hyperbolic diffeomorphisms preserving a splitting of the tangent bundle into a strong-unstable subbundleE ^uu (uniformly expanding) and a subbundleE ^c, dominated byE ^uu. We prove that if the central directionE ^c is mostly contracting for the diffeomorphism (negative Lyapunov exponents), then the ergodic Gibbsu-states are the Sinai-Ruelle-Bowen measures, there are finitely many of them, and their basins cover a full measure subset. If the strong-unstable leaves are dense, there is a unique Sinai-Ruelle-Bowen measure. We describe some applications of these results, and we also introduce a construction of robustly transitive diffeomorphisms in dimension larger than three, having no uniformly hyperbolic (neither contracting nor expanding) invariant subbundles. Work partially supported by CNRS and CNPq/PRONEX-Dynamical Systems, and carried out at Laboratoire de Topologie, Dijon, and IMPA, Rio de Janeiro.     

Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle

We prove that stable ergodicity is C ^r open and dense among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, for all r∈[2,∞]. The proof follows the Pugh–Shub program [29]: among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, accessibility is C ^r open and dense, and essential accessibility implies ergodicity. Mathematics Subject Classification (2000) Primary: 37D30, Secondary: 37A25     

Geometric expansion,Lyapunov exponents and foliations

We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological invariants and the geometric and Lyapunov growths of these foliations. As an application, we show examples of systems with persistent non-absolute continuous center and weak unstable foliations. This generalizes the remarkable results of Shub and Wilkinson to cases where the center manifolds are not compact.     

ON THE TOPOLOGICALLY CONJUGATE CLASSES OF ANOSOV ENDOMORPHISMS ON TORI

This paper considers the following question: Given an Anosov endomorphism f on T~m, whether f is topologically conjugate to some hyperbolic total endomorphism? It is well known that the answer for Anosov diffeomorphisms and expanding endomorphisms is affirmative. Hwever for the remainder Anosov endomorphisms, a quite different answer is obtained in this paper, i. e., for generic Anosov endomorphisms, they are not topologically conjugate to any hyperbolic toral endomorphism.     

Partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds and holonomy maps

In this note we show that all partially hyperbolic automorphisms on a 3-dimensional non-Abelian nilmanifold can be C¹$C^1$-approximated by structurally stable C^∞$C^{\infty }$-diffeomorphisms, whose chain recurrent set consists of one attractor and one repeller. In particular, all these partially hyperbolic automorphisms are not robustly transitive. As a corollary, the holonomy maps of the stable and unstable foliations of the approximating diffeomorphisms are twisted quasiperiodically forced circle homeomorphisms, which are transitive but non-minimal and satisfy certain fiberwise regularity properties.     

Center manifolds for nonuniformly partially hyperbolic diffeomorphisms

We establish the existence of smooth invariant center manifolds for the nonuniformly partially hyperbolic trajectories of a diffeomorphism in a Banach space. This means that the differentials of the diffeomorphism along the trajectory admit a nonuniform exponential trichotomy. We also consider the more general case of sequences of diffeomorphisms, which corresponds to a nonautonomous dynamics with discrete time. In addition, we obtain an optimal regularity for the center manifolds: if the diffeomorphisms are of class C^k then the manifolds are also of class C^k. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the center manifolds, but also for their derivatives up to order k.     

Backward inducing and exponential decay of correlations for partially hyperbolic attractors

We study partially hyperbolic attractors ofC ² diffeomorphisms on a compact manifold. For a robust (non-empty interior) class of such diffeomorphisms, we construct Sinai-Ruelle-Bowen measures, for which we prove exponential decay of correlations and the central limit theorem, in the space of Hölder continuous functions. The techniques we develop (backward inducing, redundancy elimination algorithm) should be useful in the study of the stochastic properties of much more general non-uniformly hyperbolic systems.     

How many different cascades on a surface can have coinciding hyperbolic attractors?

It is shown that the number of essentially nonconjugate (i.e., not being iterations of topologically conjugate) diffeomorphisms of a surface having homeomorphic one-dimensional hyperbolic attractors can be arbitrarily large, provided that the genus of the surface is large enough. A lower bound for this number depending on the surface genus is given. The corresponding result for pseudo-Anosov homeomorphisms is stated.     

Cocycles with one exponent over partially hyperbolic systems

We consider Hölder continuous linear cocycles over partially hyperbolic diffeomorphisms. For fiber bunched cocycles with one Lyapunov exponent we show continuity of measurable invariant conformal structures and sub-bundles. Further, we establish a continuous version of Zimmer’s Amenable Reduction Theorem. For cocycles over hyperbolic systems we also obtain polynomial growth estimates for the norm and the quasiconformal distortion from the periodic data.     

Rigidity result on conjugacies of families of diffeomorphisms

Embedding flows are used to obtain a rigidity result on strongly topological conjugacy of families of diffeomorphisms, i.e. families of C^r(2?r?∞) diffeomorphisms, the strongly topologically conjugating homeomorphisms near degenerate saddle-nodes will be differentiable on center manifolds of the saddle-nodes.     

How typical are pathological foliations in partially hyperbolic dynamics: An example

We show that for a large space of volume preserving partially hyperbolic diffeomorphisms of the 3-torus with non-compact central leaves the central foliation generically is non-absolutely continuous.     

Extremal Lyapunov exponents: an invariance principle and applications

We propose a new approach to analyzing dynamical systems that combine hyperbolic and non-hyperbolic (“center”) behavior, e.g. partially hyperbolic diffeomorphisms. A number of applications illustrate its power.