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April  2008, 2(2): 339-358. doi: 10.3934/jmd.2008.2.339

Ergodic properties of isoperimetric domains in spheres

Gerhard Knieper 1, and  Norbert Peyerimhoff 2,

1. 

Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
2. 

Department of Math. Sciences, Durham University, Durham DH1 3LE, United Kingdom

Received  September 2007 Revised  December 2007 Published  January 2008

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: spherical means, mixing., geodesic flows.
Mathematics Subject Classification: Primary 37C40, Secondary 53C12, 37C1.
Citation: Gerhard Knieper, Norbert Peyerimhoff. Ergodic properties of isoperimetric domains in spheres. Journal of Modern Dynamics, 2008, 2 (2) : 339-358. doi: 10.3934/jmd.2008.2.339
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