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Skinning measures in negative curvature and equidistribution of equidistant submanifolds

Published online by Cambridge University Press:  30 April 2013

JOUNI PARKKONEN  and
FRÉDÉRIC PAULIN
JOUNI PARKKONEN
Affiliation:
Department of Mathematics and Statistics, PO Box 35, 40014 University of Jyväskylä, Finland email [email protected]
FRÉDÉRIC PAULIN
Affiliation:
Département de mathématique, UMR 8628 CNRS, Bât. 425, Université Paris-Sud, 91405 Orsay Cedex, France email [email protected]
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Abstract

Let $C$ be a locally convex closed subset of a negatively curved Riemannian manifold $M$. We define the skinning measure ${\sigma }_{C} $ on the outer unit normal bundle to $C$ in $M$ by pulling back the Patterson–Sullivan measures at infinity, and give a finiteness result for ${\sigma }_{C} $, generalizing the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to $C$ equidistribute to the Bowen–Margulis measure ${m}_{\mathrm{BM} } $ on ${T}^{1} M$, assuming only that ${m}_{\mathrm{BM} } $ is finite and mixing for the geodesic flow. Under additional assumptions on the rate of mixing, we give a control on the rate of equidistribution.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 4 , August 2014 , pp. 1310 - 1342
Copyright
Copyright ©2013 Cambridge University Press 

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