A partial criterion for ergodicity of geodesic flows on surfaces with infinite area and negative curvature
type: Item
instance of: scholarly article
title: A partial criterion for ergodicity of geodesic flows on surfaces with infinite area and negative curvature
zbMATH Open document ID: 0796.58043
author: Hiroshi Shirakawa
published in: Proceedings of the Japan Academy. Series A
publication date: 1994-09-22T00:00:00Z
review text: If \(M=D/\Gamma\) is a surface obtained as the quotient of the Poincaré disk, \(D\), by a Fuchsian group, \(\Gamma\), then dynamic properties of the geodesic flow on \(M\) are related to properties of \(\Gamma\). Results of E. Hopf, P. J. Nicholls and the author show that the geodesic flow is ergodic (dissipative) if and only if \(M\) is of first (second) class, if and only if \(\Gamma\) is of divergence (convergence) type [the author: Comment. Math. Univ. St. Pauli 31, 163-182 (1982; Zbl 0506.58027)]. In this paper, surfaces with infinite area and with variable negative curvature are investigated. It is proved that if the set of points, where the metric is not constant, is a set with compact closure, and if the directional derivative of the Gaussian curvature along geodesic circles is uniformly bounded, then the result stated above is again valid.
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 559493
zbMATH Keywords: geodesic flows
zbMATH Keywords: Fuchsian group
zbMATH Keywords: Poincaré metric
reviewed by: D. J. Hurley
MaRDI profile type: Publication
full work available at URL: pjaa.69.115
OpenAlex ID: W2067859945
cites work: Surfaces of negative curvature and permanent regional transitivity
cites work: The measure of geodesic types on surfaces of negative curvature
cites work: Q5778153
cites work: Statistik der Lösungen geodätischer Probleme vom unstabilen Typus. 2
cites work: Transitivity Properties of Fuchsian Groups
cites work: Q4743476
DOI: 10.3792/PJAA.69.115
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description: scientific article; zbMATH DE number 559493