Journal of Modern Dynamics (JMD)

Ergodic properties of isoperimetric domains in spheres

Pages: 339 - 358, Issue 2, April 2008      doi:10.3934/jmd.2008.2.339

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Gerhard Knieper - Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany (email)
Norbert Peyerimhoff - Department of Math. Sciences, Durham University, Durham DH1 3LE, United Kingdom (email)

Abstract: Let $\varphi$ be a function on the unit tangent bundle of a compact manifold of negative curvature. We show that averages of $\varphi$ over subdomains of increasing spheres converge to the horospherical mean if these domains satisfy an isoperimetric condition. We apply this result to spherical means with continuous density and, by using relations between the horospherical mean and the Patterson-Sullivan measure, we derive some kind of mixing properties.

Keywords:  geodesic flows, spherical means, mixing.
Mathematics Subject Classification:  Primary 37C40, Secondary 53C12, 37C10.

Received: September 2007;      Revised: December 2007;      Available Online: January 2008.