Recurrence statistics for the space of Interval Exchange maps and the Teichmüller flow on the space of translation surfaces

Abstract: In this note we show that the transfer operator of a Rauzy-Veech-Zorich renormalization map acting on a space of quasi-H\"older functions is quasicompact and derive certain statistical recurrence properties for this map and its associated Teichm\"uller flow. We establish Borel-Cantelli lemmas, Extreme Value statistics and return time statistics for the map and flow. Previous results have established quasicompactness in H\"older or analytic function spaces, for example the work of M. Pollicott and T. Morita. The quasi-H\"older function space is particularly useful for investigating return time statistics. In particular we establish the shrinking target property for nested balls in the setting of Teichm\"uller flow. Our point of view, approach and terminology derives from the work of M. Pollicott augmented by that of M. Viana.