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Counting closed geodesics in a compact rank-one locally CAT(0) space

Part of: Global differential geometry Measure-theoretic ergodic theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  27 August 2021

RUSSELL RICKS
RUSSELL RICKS*
Affiliation:
Binghamton University, Binghamton, New York, USA
*
e-mail: ricks@math.binghamton.edu
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Abstract

Let X be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank-one axis. Assume X is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of oriented closed geodesics of length at most t; then $\lim \nolimits _{t \to \infty } P_t / ({e^{ht}}/{ht}) = 1$ , where h is the entropy of the geodesic flow on the space $GX$ of parametrized unit-speed geodesics in X.

Keywords

classical ergodic theoryentropynon-positive curvature

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53C22: Geodesics
Secondary: 28D20: Entropy and other invariants
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 1220 - 1251
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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