On the Ergodicity of Flat Surfaces of Finite Area

Abstract

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.

23 pages. Accepted version to appear in GAFA

References in corpus (2)

• Non-ergodic Z-periodic billiards and infinite translation surfaces
• Ergodic infinite group extensions of geodesic flows on translation surfaces

Cited by in corpus (7)

• Flat surface models of ergodic systems
• Wild translation surfaces and infinite genus
• Indiscriminate covers of infinite translation surfaces are innocent, not devious
• Logarithmic laws and unique ergodicity
• Effective Unique Ergodicity and Weak Mixing of Translation Flows
• Rotational component spaces for infinite-type translation surfaces
• Flat surfaces, Bratteli diagrams, and unique ergodicity à la Masur