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Abstract

For a non-compact hyperbolic surface $M$ of finite area, we study a certain Poincaré section for the geodesic flow. The canonical, non-invertible factor of the first return map to this section is shown to be pointwise dual ergodic with return sequence $(a_n)$ given by $$ a_n = \frac{\pi}{ 4 (\hbox{Area}(M) + 2 \pi)}\cdot \frac{n}{\log n }. $$ We use this result to deduce that the section map itself is rationally ergodic, and that the geodesic flow associated to $M$ is ergodic with respect to the Liouville measure.