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On the ergodicity of geodesic flows on surfaces without focal points
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 12 / December 2023
- Published online by Cambridge University Press:
- 03 February 2023, pp. 4226-4248
- Print publication:
- December 2023
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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.
