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 (aₙ) given by aₙ = 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-3-3,
author = {Manuel Stadlbauer},
title = {The return sequence of the Bowen-Series map for punctured surfaces},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {221-240},
zbl = {1095.37008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-3-3}
}
Manuel Stadlbauer. The return sequence of the Bowen-Series map for punctured surfaces. Fundamenta Mathematicae, Tome 184 (2004) pp. 221-240. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-3-3/