The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center

We prove that if f is a partially hyperbolic diffeomorphism on the compact manifold M with one-dimensional center bundle, then the logarithm of the spectral radius of the map induced by f on the real homology groups of M is smaller or equal to the topological entropy of f. This is a particular case of the Shub's entropy conjecture, which claims that the same conclusion should be true for any C¹ map on any compact manifold.     

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.     

Accessibility of partially hyperbolic endomorphisms with 1D center-bundles

We prove that partially hyperbolic endomorphisms with one dimensional center-bundles and non-trivial unstable bundles are stably accessible. And there is residual subset $\Res$ of partially hyperbolic volume preserving endomorphisms with one dimensional center-bundles such that every $f \in \Res$ is stably accessible. In the end, we prove the accessibility of Gan''s example.     

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...     

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.     

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.     

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.     

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.     

Horseshoes for diffeomorphisms preserving hyperbolic measures

We give extensions of Katok’s horseshoe constructions, comment on related results, and provide a self-contained proof. We consider either a \(C^{1+\alpha }\) diffeomorphism preserving a hyperbolic measure or a \(C^1\) diffeomorphism preserving a hyperbolic measure whose support admits a dominated splitting.     

Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle

We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are $C^r$ open and there exists a $C^r$ open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.     

Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center To the Memory of Professor Shantao Liao

In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive partially hyperbolic diffeomorphisms on 3-manifolds with topologically neutral center.     

Cubic tangencies and hyperbolic diffeomorphisms

We show that the loss of hyperbolicity of an Anosov diffeomorphism of the torusT ² can be produced by a cubic tangency at a heteroclinic point. Such a first bifurcation is generic for 3-parameters families of diffeomorphisms. Our construction may also be applied to any basic set of a surface diffeomorphism. Moreover, if the pointq of cubic tangency corresponds to a lateral point of then the bifurcation is generic for two parameters. In this case the pointq may be a homoclinic intersection.Dedicated to the memory of R. MañéPartially supported by CNPq (Brazil) and CNRS (France).Supported by CNRS (France), Rectorat Université de Bourgogne (France) and CNPq (Brazil).     

On local ergodicity in hyperbolic systems with singularities

United Institute of Nuclear Research. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 1, pp. 60–64, January–March, 1993.     

Quantum hyperbolic invariants for diffeomorphisms of small surfaces

An earlier article [Bonahon, F., Liu, X. B.: Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol., 11, 889-937 (2007)] introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmu¨ller space. We explicitly compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.     

Equality of pressures for diffeomorphisms preserving hyperbolic measures

For a diffeomorphism which preserves a hyperbolic measure the potential is studied. Various types of pressure of are introduced. It is shown that these pressures satisfy a corresponding variational principle. This research was supported by the grant EU FP6 ToK SPADE2. The author is grateful to IM PAN Warsaw for the hospitality and to C. Wolf for discussions about suitable concepts of pressure.