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1 June 2021 Local limit theorem in negative curvature
François Ledrappier, Seonhee Lim
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Duke Math. J. 170(8): 1585-1681 (1 June 2021). DOI: 10.1215/00127094-2020-0069

Abstract

Consider the heat kernel (t,x,y) on the universal cover M˜ of a closed Riemannian manifold of negative sectional curvature. We show the local limit theorem for :

limtt32eλ0t(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 λ0-harmonic function which depends on x,yM˜. We also show that the λ0-Martin boundary of M˜ is equal to its topological boundary. The Martin decomposition of C(x,y) gives a family of measures {μxλ0} on M˜. We show that {μxλ0} is a family minimizing the energy or Mohsen’s Rayleigh quotient. We apply the uniform Harnack inequality on the boundary M˜ and the uniform three-mixing of the geodesic flow on the unit tangent bundle SM for suitable Gibbs–Margulis measures.

Citation

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François Ledrappier. Seonhee Lim. "Local limit theorem in negative curvature." Duke Math. J. 170 (8) 1585 - 1681, 1 June 2021. https://doi.org/10.1215/00127094-2020-0069

Information

Received: 10 June 2018; Revised: 5 September 2020; Published: 1 June 2021
First available in Project Euclid: 18 January 2021

Digital Object Identifier: 10.1215/00127094-2020-0069

Subjects:
Primary: 37D40
Secondary: 37A17, 37A25, 37A30, 37A50

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 8 • 1 June 2021
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