Ergodic properties of isoperimetric domains in spheres

Gerhard Knieper; Norbert Peyerimhoff

doi:10.3934/jmd.2008.2.339

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2008, Volume 2, Issue 2: 339-358. Doi: 10.3934/jmd.2008.2.339

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Ergodic properties of isoperimetric domains in spheres

Gerhard Knieper^1, and
Norbert Peyerimhoff^2,

1.
Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum
2.
Department of Math. Sciences, Durham University, Durham DH1 3LE

Received: August 31, 2007

Revised: November 30, 2007

Published: December 2007

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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.

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Mathematics Subject Classification: Primary 37C40, Secondary 53C12, 37C10.

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