Pseudo-anosov Foliations on Periodic Surfaces

In this note we shall study the lifts of stable foliations of pseudo-Anosov diffeomorphism to certain infinite abelian covers. This is motivated, at least in part, by recent progress in understanding the ergodic properties of the analogous horocycle flows on infinite surfaces [3],[13]. Our aim is to show that many of the results from that context hold in this natural and technically simpler setting. It has long been appreciated that the stable foliation of an Anosov diffeomorphism of a compact manifold and the horocycle foliation of the unit tangent bundle of a compact negatively curved surface have many similar properties, although in the former case the technical analysis is often more straightforward. In particular, it is a classical fact that the horocycle foliation is uniquely ergodic [10] and, as shown by Bowen and Marcus, this property also holds for the stable foliation of an Anosov (or Axiom A) diffeomorphism [6]. Furthermore, this property extends to the broader class of pseudo-Anosov maps of surfaces. Let us recall the following result of Thurston.