A partial criterion for ergodicity of geodesic flows on surfaces with infinite area and negative curvature
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.
scientific article; zbMATH DE number 559493
scientific article