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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 = π 4(Area(M ) + 2π) · 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.doi:10.4064/fm182-3-3 fatcat:zvej7jehf5c3hbqihyhrjba3au