Related papers: On the ergodicity of partially hyperbolic systems
We show that certain types of the three-legged accessibility property of a partially hyperbolic diffeomorphism imply the existence of a unique minimal set for one strong foliation and the transitivity of the other one. In case the center…
We consider Holder 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…
In this paper we continue the analysis of non-diagonalisable hyperbolic systems initiated in \cite{GarJRuz, GarJRuz2}. Here we assume that the system has discontinuous coefficients or more in general distributional coefficients.…
Every volume-preserving centre-bunched fibred partially hyperbolic system with 2-dimensional centre either (1) has two distinct centre Lyapunov exponents, or (2) exhibits an invariant continuous line field (or pair of line fields) tangent…
We study nonhyperbolic and transitive partially hyperbolic diffeomorphisms having a one-dimensional center. We prove joint flexibility with respect to entropy and center Lyapunov exponent for a broad class of these systems. Flexibility…
We study fibered partially hyperbolic diffeomorphisms. We show that as long as certain topological obstructions vanish and as long as homological minimum expansion dominates the distortion on the fibers that a fibered partially hyperbolic…
We call a partially hyperbolic diffeomorphism \emph{partially volume expanding} if the Jacobian restricted to any hyperplane that contains the unstable bundle $E^u$ is larger than $1$. This is a $C^1$ open property. We show that any…
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.
We show that any conservative partially hyperbolic diffeomorphism homotopic to the identity is accessible unless the fundamental group of its ambient 3-manifold is virtually solvable. As a consequence, such diffeomorphisms are ergodic,…
Under a suitable bunching condition, we establish that stable holonomies inside center-stable manifolds for $C^{1+\beta}$ diffeomorphisms are uniformly bi-Lipschitz and in fact $C^{1+\text{H\"older}}$. This verifies that the Pugh-Shub…
In [Discrete Contin. Dyn. Syst. \textbf{15} (2006), no. 3, 811--818.] Xia introduced a simple dynamical density basis for partially hyperbolic sets of volume preserving diffeomorphisms. We apply the density basis to the study of the…
It is one of the main properties of uniformly hyperbolic dynamics that points of two distinct trajectories cannot be uniformly close one to another. This characteristics of hyperbolic dynamics is called expansivity. Hirsch, Pugh and Shub,…
We give explicit $C^1$-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with…
We use the Invariance Principle of Avila and Viana to prove that every partially hyperbolic symplectic diffeomorphism with 2-dimensional center bundle, and satisfying certain pinching and bunching conditions, can be $C^r$-approximated by…
In this paper we improve the results of \cite{MT} and show that a weak hyperbolic transitivity implies the uniqueness of hyperbolic SRB measures. As an important corollary, it arises the ergodicity of the system in a conservative setting.…
Some of the guiding problems in partially hyperbolic systems are the following: (1) Examples, (2) Properties of invariant foliations, (3) Accessibility, (4) Ergodicity, (5) Lyapunov exponents, (6) Integrability of central foliations, (7)…
We prove a sharp phase transition in the regularity of the extremal distribution $E^s \oplus E^u$ for $C^\infty$ volume-preserving partially hyperbolic diffeomorphisms on closed $3$-manifolds: if $E^s \oplus E^u$ is Lipschitz, then it is…
The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper…
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,…
We prove the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms where the center direction has a dominated decomposition into one dimensional bundle and there is a uniform lower bound for the absolute…