Homology of closed geodesics in a negatively curved manifold
type: Item
instance of: scholarly article
title: Homology of closed geodesics in a negatively curved manifold
zbMATH Open document ID: 0618.58028
author: Toshiaki Adachi
author: Toshikazu Sunada
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: \textit{W. Klingenberg} has shown [Ann. Math., II. Ser. 99, 1-13 (1974; Zbl 0272.53025)] that every compact manifold M whose geodesic flow on the unit tangent bundle is Anosov has a unique geodesic in every free homotopy class of closed paths. The authors use combinatorial arguments and Ruelle's infinite dimensional Frobenius-Perron theorem [\textit{D. Ruelle}, Thermodynamic formalism (1978; Zbl 0401.28016)] to prove that there are infinitely many geodesics with primitive period in each homology class in \(H_ 1(M,Z)\) in the sense that for the number N(x,\(\alpha)\) of such geodesics of given homology class \(\alpha\) and length \(\leq x\) we have \(\lim_{x\to \infty} x^{-1} \log N(x,\alpha)=the\) (positive) topological entropy of the geodesic flow.
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 4003113
zbMATH Keywords: geodesic flow
zbMATH Keywords: Anosov
Wikidata QID: Q115184729
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214441177
OpenAlex ID: W1506261457
DOI: 10.4310/JDG/1214441177
description: scientific article; zbMATH DE number 4003113