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In this note we show that the transfer operator of a Rauzy-Veech-Zorichrenormalization map acting on a space of quasi-H\"older functions isquasicompact and derive certain statistical recurrence properties for this mapand 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 analyticfunction spaces, for example the work of M. Pollicott and T. Morita. Thequasi-H\"older function space is particularly useful for investigating returntime statistics. In particular we establish the shrinking target property fornested balls in the setting of Teichm\"uller flow. Our point of view, approachand terminology derives from the work of M. Pollicott augmented by that of M.Viana.

Recurrence statistics for the space of Interval Exchange maps and the Teichm\"uller flow on the space of translation surfaces