• DOI:10.1007/S00039-014-0269-4
• Corpus ID: 18850181

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}
}

• Rodrigo Treviño
• Published 6 November 2012
• Mathematics
• Geometric and Functional Analysis

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 criterion for ergodicity based on the control of deforming metric of a flat surface. Applied to translation flows on compact surfaces, it improves and generalizes a theorem of Cheung and Eskin et al. (Partially hyperbolic dynamics, laminations, and Teichmüller flow, Fields Inst. Commun., Vol. 51. Amer. Math. Soc., Providence, pp. 213–221, 2007).