Abstract
We prove the equidistribution of (weighted) periodic orbits of the geodesic ow on noncompact negatively curved manifolds toward equilibrium states in the narrow topology, i.e. in the dual of bounded continuous functions. We deduce an exact asymptotic counting for periodic orbits (weighted or not), which was previously known only for geometrically finite manifolds.
Highlights
A well known feature of compact hyperbolic dynamics is the abundance of periodic orbits: they have a positive exponential growth rate, equal to the topological entropy
Let M be a manifold with pinched negative curvature, whose geodesic ow is topologically mixing
Given p ∈ PWR (T − c, T ) \ PW (T − c, T ), choose arbitrarily one isometry γp ∈ Γ, whose translation axis intersects WR \ W and projects on M on the closed geodesic associated to the periodic orbit p, and whose translation length is (p)
Summary
A well known feature of compact hyperbolic dynamics is the abundance of periodic orbits: they have a positive exponential growth rate, equal to the topological entropy. A weighted version of this result holds : given a Hölder continuous map F : T 1M → R, as soon as it admits a nite equilibrium state mF the orbital measures supported by periodic orbits of length at most T , conveniently weighted by the periods of the potential F , converge to the normalized measure m F in the vague topology. It turns out that the equidistribution property required to get such counting estimates is a stronger convergence, in the narrow topology, i.e. the dual of continuous bounded functions Until now, such narrow equidistribution or such asymptotic counting for periodic orbits have been proven only when M is geometrically nite in [Rob[03], PPS15].
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