Equidistribution of horospheres in nonpositive curvature

DOI10.1017/ETDS.2021.124zbMATH Open1516.37005arXiv2007.05253WikidataQ114118998 ScholiaQ114118998MaRDI QIDQ6344812

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Publication date: 10 July 2020

Abstract: We study the ergodic properties of horospheres on rank 1 manifolds with non-positive curvature. We prove that the horospheres are equidistributed under the action of the geodesic flow towards the Bowen-Margulis measure, on a large class of manifolds. In the case of surfaces, we define a parametrization of the horocyclic flow on the set of horocycles containing a rank 1 vector that is recurrent under the action of the geodesic flow. We prove that the horocyclic flow in restriction to this set is uniquely ergodic. The results are valid for large classes of manifolds, including the compact ones.

Mathematics Subject Classification ID

Ergodicity, mixing, rates of mixing (37A25) Geodesic flows in symplectic geometry and contact geometry (53D25) Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) (37D40) Homogeneous flows (37A17)

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