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- Title
Volume asymptotics, Margulis function and rigidity beyond nonpositive curvature.
- Authors
Wu, Weisheng
- Abstract
In this article, we consider a closed rank one C ∞ Riemannian manifold M without focal points and its universal cover X. Let b t (x) be the Riemannian volume of the ball of radius t > 0 around x ∈ X , and h the topological entropy of the geodesic flow. We obtain the following Margulis-type asymptotic estimates lim t → ∞ b t (x) / e ht h = c (x) for some continuous function c : X → R. We prove that the Margulis function c(x) is in fact C 1. The result also holds for a class of manifolds without conjugate points, including all surfaces of genus at least 2 without conjugate points. If M is a rank one surface without focal points, we show that c(x) is constant if and only if M has constant negative curvature. We also obtain a rigidity result related to the flip invariance of the Patterson–Sullivan measure. These rigidity results are new even in the nonpositive curvature case.
- Subjects
GEODESIC flows; CURVATURE; TOPOLOGICAL entropy; CONTINUOUS functions; RIEMANNIAN manifolds
- Publication
Mathematische Annalen, 2024, Vol 389, Issue 3, p2317
- ISSN
0025-5831
- Publication type
Article
- DOI
10.1007/s00208-023-02710-x