Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center
Keith Burns Federico Rodriguez Hertz María Alejandra Rodriguez Hertz Anna Talitskaya Raúl Ures
It is shown that stable accessibility property is $C^r$-dense among partially hyperbolic diffeomorphisms with one-dimensional center bundle, for $r \geq 2$, volume preserving or not. This establishes a conjecture by Pugh and Shub for these systems.
keywords: density of accessibility one dimensional center. partial hyperbolicity
Dynamical coherence and center bunching
Keith Burns Amie Wilkinson
This paper discusses relationships among the basic notions that have been important in recent investigations of the ergodicity of volume-preserving partially hyperbolic diffeomorphisms. In particular we survey the possible definitions of dynamical coherence and discuss the relationship between dynamical coherence and center bunching.
keywords: partial hyperbolicity stable ergodicity. dynamical coherence center bunching
Lyapunov spectrum for geodesic flows of rank 1 surfaces
Keith Burns Katrin Gelfert
We give estimates on the Hausdorff dimension of the levels sets of the Lyapunov exponent for the geodesic flow of a compact rank 1 surface. We show that the level sets of points with small (but non-zero) exponents has full Hausdorff dimension, but carries small topological entropy.
keywords: Lyapunov exponents geodesic flow rank 1 surfaces Hausdorff dimension shadowing. entropy pressure multifractal formalism
Growth of the number of geodesics between points and insecurity for Riemannian manifolds
Keith Burns Eugene Gutkin
A Riemannian manifold is said to be uniformly secure if there is a finite number $s$ such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by $s$ point obstacles. We prove that the number of geodesics with length $\leq T$ between every pair of points in a uniformly secure manifold grows polynomially as $T \to \infty$. By results of Gromov and Mañé, the fundamental group of such a manifold is virtually nilpotent, and the topological entropy of its geodesic flow is zero. Furthermore, if a uniformly secure manifold has no conjugate points, then it is flat. This follows from the virtual nilpotency of its fundamental group either via the theorems of Croke-Schroeder and Burago-Ivanov, or by more recent work of Lebedeva.
    We derive from this that a compact Riemannian manifold with no conjugate points whose geodesic flow has positive topological entropy is totally insecure: the geodesics between any pair of points cannot be blocked by a finite number of point obstacles.
keywords: entropy security connecting geodesics.
Stable ergodicity for partially hyperbolic attractors with negative central exponents
Keith Burns Dmitry Dolgopyat Yakov Pesin Mark Pollicott
We establish stable ergodicity of diffeomorphisms with partially hyperbolic attractors whose Lyapunov exponents along the central direction are all negative with respect to invariant SRB-measures.
keywords: Partial hyperbolicity stable ergodicity accessibility Lyapunov exponents SRB-measures.

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