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arXiv · 1812.04409

On the ergodicity of geodesic flows on surfaces without focal points

Weisheng Wu·Fei Liu·Fang Wang
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Abstract

In this article, 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.

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Weisheng Wu, Fei Liu, Fang Wang. 2018-12-11. On the ergodicity of geodesic flows on surfaces without focal points. https://arxiv.org/abs/1812.04409

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