Growth of the number of geodesics between points and insecurity for Riemannian manifolds
Keith Burns Eugene Gutkin
Discrete & Continuous Dynamical Systems - A 2008, 21(2): 403-413 doi: 10.3934/dcds.2008.21.403
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
Journal of Modern Dynamics 2008, 2(1): 63-81 doi: 10.3934/jmd.2008.2.63
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.
Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center
Keith Burns Federico Rodriguez Hertz María Alejandra Rodriguez Hertz Anna Talitskaya Raúl Ures
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 75-88 doi: 10.3934/dcds.2008.22.75
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
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 89-100 doi: 10.3934/dcds.2008.22.89
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
Discrete & Continuous Dynamical Systems - A 2014, 34(5): 1841-1872 doi: 10.3934/dcds.2014.34.1841
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

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