# On the ergodicity of flat surfaces of finite area

@article{Trevio2012OnTE, title={On the ergodicity of flat surfaces of finite area}, author={Rodrigo Trevi{\~n}o}, journal={Geometric and Functional Analysis}, year={2012}, volume={24}, pages={360-386} }

We prove some ergodic theorems for flat surfaces of finite area. The first result concerns such surfaces whose Teichmüller orbits are recurrent to a compact set of $${SL(2,\mathbb{R})/SL(S,\alpha)}$$SL(2,R)/SL(S,α) , where SL(S,α) is the Veech group of the surface. In this setting, this means that the translation flow on a flat surface can be renormalized through its Veech group. This result applies in particular to flat surfaces of infinite genus and finite area. Our second result is an…

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