Abstract
AbstractWe 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.
Highlights
Horocyclic flows associated to a geodesic flow have been extensively studied on compact surfaces with constant negative curvature [10], and later on compact surfaces with variable negative curvature [15, 16]
Babillot gave a simple proof of the mixing property of the geodesic flow and showed the equidistribution of horospheres under the action of this flow towards certain product measures for manifolds with negative curvature [1]
Thanks to the equidistribution of the horocycles of Theorem A and a part of the strategy followed by Coudène in [4], we prove the unique ergodicity of the horocyclic flow on for manifolds that satisfy the duality condition, which means that the non-wandering set of the geodesic flow is the whole unitary tangent bundle
Summary
Horocyclic flows associated to a geodesic flow have been extensively studied on compact surfaces with constant negative curvature [10], and later on compact surfaces with variable negative curvature [15, 16]. For every unstable horosphere H ⊂ T 1M, every open subset U of H containing a non-wandering vector is equidistributed under the action of the geodesic flow; that is, for every bounded and uniformly continuous function f on T 1M, we have
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