On the ergodicity of geodesic flows on surfaces without focal points

TL;DR

This paper proves the ergodicity of geodesic flows on certain closed surfaces without focal points, under specific curvature conditions, extending understanding of dynamical systems on such geometric structures.

Contribution

It establishes ergodicity of geodesic flows on surfaces with no focal points and finitely many negatively curved regions, a significant extension in geometric dynamical systems.

Findings

Geodesic flow is ergodic on the unit tangent bundle.

Ergodicity holds under the condition of finitely many negatively curved components.

The result applies to surfaces with no focal points and genus at least 2.

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.