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On the ergodicity of geodesic flows on surfaces without focal points

Part of: Dynamical systems with hyperbolic behavior Ergodic theory

Published online by Cambridge University Press:  03 February 2023

WEISHENG WU ,
FEI LIU  and
FANG WANG
WEISHENG WU*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China
FEI LIU
Affiliation:
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China (e-mail: feiliu.sdust@gmail.com)
FANG WANG
Affiliation:
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Beijing Center for Mathematics and Information Interdisciplinary Sciences (BCMIIS), Beijing 100048, P. R. China (e-mail: fangwang@cnu.edu.cn)
*
e-mail: wuweisheng@xmu.edu.cn
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Abstract

In this paper, we study the ergodicity of the geodesic flows on surfaces with no focal points. Let M be a smooth connected and closed surface equipped with a $C^{\infty }$ Riemannian metric g, whose genus $\mathfrak {g} \geq 2$. Suppose that $(M,g)$ has no focal points. We prove that the geodesic flow on the unit tangent bundle of M is ergodic with respect to the Liouville measure, under the assumption that the set of points on M with negative curvature has at most finitely many connected components.

Keywords

ergodicitygeodesic flowno focal pointsnon-uniform hyperbolicity

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 37A25: Ergodicity, mixing, rates of mixing 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 12 , December 2023 , pp. 4226 - 4248
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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