Invariant measures for the linear flow on infinite translation surfaces

Translation surfaces are a class of locally flat surfaces that can be obtained by gluing together edges of a collection of polygons. There is a natural flow that can be defined on a translation surface, the straight-line flow in a given direction. The dynamics of this flow is very well understood for compact translation surfaces, with classical results telling us that for almost all directions, Lebesgue is the unique invariant measure for the flow.

For translation surfaces of infinite type however the picture is very different, and very little is still known about the classification of invariant measures of the linear flow. I will talk about the simplest case, where the infinite type surface is the Z^d cover of a compact translation surface and further the infinite surface is periodic under a renormalisation map. In this case it turns out that one can classify the ergodic invariant measures and even obtain a somewhat explicit form for them.