The return sequence of the Bowen–Series map for punctured surfaces

Manuel Stadlbauer ¹

¹ Institut für Mathematische Stochastik Maschmühlenweg 8–10 37073 Göttingen, Germany

Fundamenta Mathematicae, Tome 182 (2004) no. 3, pp. 221-240

Résumé

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.

Détail

DOI : 10.4064/fm182-3-3

Keywords: non compact hyperbolic surface finite area study certain poincar section geodesic flow canonical non invertible factor first return map section shown pointwise dual ergodic return sequence given frac hbox area cdot frac log result deduce section map itself rationally ergodic geodesic flow associated ergodic respect liouville measure