Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

$\mathbb Z^d$-covers of horosphere foliations

Pages: 147 - 154, Volume 6, Issue 1, January 2000      doi:10.3934/dcds.2000.6.147

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Mark Pollicott - Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom (email)

Abstract: 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:  Ergodicity, horospheres, foliations, geodesic flow, symbolic dynamics.
Mathematics Subject Classification:  Primary: 28D05, 58F11.

Received: October 1999;      Published: December 1999.