Local Limit Theorem in negative curvature

2015/03/13 by Ledrappier, François, Lim, Seonhee · 3 citations
#37A17 #37A25 #37A30 #37A50 #37D40 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1503.04156

Abstract

Consider the heat kernel p(t,x,y) on the universal cover X of a Riemannian manifold M of negative curvature. We show the local limit theorem for p : limt → ∞ t3/2eλ0 t p(t,x,y)=C(x,y), where λ0 is the bottom of the spectrum of the geometric Laplacian and C(x,y) is a positive function which depends on x, y ∈ X. We also show that the λ0-Martin boundary of X is equal to its topological boundary. The Martin decomposition of C(x,y) gives a family of measures \μλ0x \ on ∂ \widetildeM. We show that \μλ0x \ is the unique family minimizing the energy or the Rayleigh quotient of Mohsen. We use the uniform Harnack inequality on the boundary ∂ X and the uniform three-mixing of the geodesic flow on the unit tangent bundle SM for suitable Gibbs-Margulis measures.

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