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Skinning measures in negative curvature and equidistribution of equidistant submanifolds
Published online by Cambridge University Press: 30 April 2013
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.
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- Copyright ©2013 Cambridge University Press
References
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