Skinning measures in negative curvature and equidistribution of equidistant submanifolds

Abstract

Let C be a locally convex subset of a negatively curved Riemannian manifold M . We define the skinning measure σ 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 of σ C , generalising 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 BM on T 1 M , assuming only m 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. 1

A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature

“…We refer for instance to [Rob2,PaP3,PaPS,BrPP] for definitions, proofs and complements concerning this subsection. We introduce here the various measures that will be useful for our ergodic study in Section 4…”

“…The inner (respectively outer) unit normal bundle ∂ 1 − C (respectively ∂ 1 + C) of C is the topological submanifold of T 1 M consisting of the unit tangent vectors v ∈ T 1 M such that π(v) ∈ ∂C, v is orthogonal to a contact hyperplane to C and points towards (respectively away from) C (see [PaP3] for more precisions, and note that the boundary ∂C of C is not necessarily C 1 , hence may have more than one contact hyperplane at some point, and that it is not necessarily true that exp(tv) belongs (respectively does not belong) to C for t > 0 small enough). 000000 000000 000000 111111 111111 111111…”

“…Using this homeomorphism, we defined in [PaP3] the outer (respectively inner) skinning measure on T 1 M as the measure σ + C (respectively σ − C ) with support in ∂ 1 + C (respectively ∂ 1 − C) given by, for all w ∈ ∂ 1 ± C,…”

On the hyperbolic orbital counting problem in conjugacy classes

“….2] when D ∓ is a horoball or a totally geodesic subspace in M and [PP2], [PP4] for the general case of convex subsets in variable curvature and with a potential.…”

“…We refer for instance to [Rob2] for the definition of the critical exponent δ Γ of Γ, the Patterson-Sullivan measures (µ x ) x∈H 2 R of Γ, the Bowen-Margulis measure m BM of Γ, and to [OhS1,PP2] for the definition of the skinning measure σ C of Γ associated to a nonempty proper closed convex subset C of H 2 R (see also Section 3). We denote by µ the total mass of a measure µ and by ∆ x the unit Dirac mass at a point x.…”