$\mathbb Z^d$-covers of horosphere foliations
Mark Pollicott
Discrete & Continuous Dynamical Systems - A 2000, 6(1): 147-154 doi: 10.3934/dcds.2000.6.147
Let $M$ be the unit tangent bundle of a compact manifold with negative sectional curvatures and let $\hat M$ be a $\mathbb Z^d$ cover for $M$. Let $\mu$ be the measure of maximal entropy for the associated geodesic flow on $M$ and let $\hat\mu$ be the lift of $\mu$ to $\hat M$.
We show that the foliation $\hat{M^{s s}}$ is ergodic with respect to $\hat\mu$. (This was proved in the special case of surfaces by Babillot and Ledrappier by a different method.) Our method extends to certain Anosov and hyperbolic flows.
keywords: foliations geodesic flow symbolic dynamics. horospheres Ergodicity
The Hausdorff dimension of measures for iterated function systems which contract on average
Thomas Jordan Mark Pollicott
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 235-246 doi: 10.3934/dcds.2008.22.235
In this note we consider measures supported on limit sets of systems that contract on average. In particular, we present an upper bound on their Hausdorff dimension.
keywords: Contraction on average Iterated function scheme Hausdorff dimension
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.
Closed geodesic distribution for manifolds of non-positive curvature
Mark Pollicott
Discrete & Continuous Dynamical Systems - A 1996, 2(2): 153-161 doi: 10.3934/dcds.1996.2.153
In this paper we give a generalization of Bowen's equidistribution result for closed geodesics on negatively curved manifolds to rank one manifolds.
keywords: Bowen's equidistribution closed geodesics.
Local Hölder regularity of densities and Livsic theorems for non-uniformly hyperbolic diffeomorphisms
Mark Pollicott
Discrete & Continuous Dynamical Systems - A 2005, 13(5): 1247-1256 doi: 10.3934/dcds.2005.13.1247
We consider the partial analogue of the usual measurable Livsic theorem for Anosov diffeomorphims in the context of non-uniformly hyperbolic diffeomorphisms (Theorem 2). Our main application of this theorem is to the density of absolutely continuous measures (Theorem 1).
keywords: Hölder function. Anosov flows stable manifolds
Closed orbits and homology for $C^2$-flows
Mark Pollicott
Discrete & Continuous Dynamical Systems - A 1999, 5(3): 529-534 doi: 10.3934/dcds.1999.5.529
keywords: de Rham cohomology. Manifold cohomology probability measure winding cycle
Ergodicity of stable manifolds for nilpotent extensions of Anosov flows
Mark Pollicott
Discrete & Continuous Dynamical Systems - A 2002, 8(3): 599-604 doi: 10.3934/dcds.2002.8.599
In this paper we show ergodicity of the strong stable foliations for nilpotent extensions of transitive Anosov flows with respect to the lift of the Gibbs measure for any Hölder continuous function.
keywords: Anosov flows Hölder function. Stable manifolds

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