  We prove some ergodic theorems for flat surfaces of finite area. The first
result concerns such surfaces whose Teichmuller orbits are recurrent to a
compact subset of $SL(2;R)/SL(S)$, where $SL(S)$ 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 to reveal ergodic
properties of the translation flow. 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.
