On the Ergodicity of Flat Surfaces of Finite Area

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,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 a criterion for unique ergodicity for translation flows on compact surfaces which improves and generalizes a theorem of Cheung and Eskin[CE07]. A flat surface is a two-dimensional oriented manifold S endowed with a flat metric everywhere except on a set of “bad points” Σ or singularities, which are forced to exist by the topology of the surface if the surface is of genus greater than one. Flat surfaces are inextricably connected to quadratic differentials since the latter give a Riemann surface a flat metric and a pair of transverse, measured foliations, called the vertical and horizontal foliations. If the foliations are orientable, which is not always the case, by considering them as flows we suddenly have a dynamical system, called the translation flow. In other words, since the foliations are orientable, the holomorphic 1-form or an Abelian differential α defining the quadratic differential defines a dynamical system on a surface. Thus one can try to derive dynamical and ergodic properties of the flow by studying properties of the Abelian differential. Although we can get two different flows by considering the horizontal or vertical foliations, from now on we shall assume the flow corresponds to the horizontal foliation, which can be thought as being defined along a global direction θα ∈ S. This flow preserves an absolutely continuous measure ωα, singular at Σ, which is also defined by the Abelian differential. For a very thorough background on flat surfaces, see [MT02, Zor06]. In the case when the surface is compact, the point of view of looking at a quadratic differential in order to derive properties of the dynamical system which it defines is a rather favorable one, as the “right” space of all quadratic differentials on a fixed Riemann surface of genus g is a finite dimensional space. This “right” space is the moduli space of quadratic differentials, or moduli space for short. It is the “right” space because it has many convenient properties. For example, it is the space of classes of conformally-equivalent flat metrics on a Riemann surface, it is a topological space homeomorphic to an open ball of dimension 6g − 6 (where g is the genus of the surface), and it is equipped with an absolutely continuous SL(2,R)-invariant probability measure [Mas82, Vee86]. Properties of the translation flow on a compact flat surface can be derived from an associated dynamical Date: October 4, 2013.