Mather measures and the Bowen-Series transformation. (Q850171)

The paper studies a particular compact Riemann surface \(M\) of genus 2 and constant negative curvature. A ``geodesic billiard'' is defined in terms of the geodesic flow on the unit tangent bundle. An explicit description of \(M\) as a quotient space of a specific octagonal domain in the hyperbolic disk is used to define a Bowen-Series transformation \(T\) of the circle at infinity, and a natural extension \(\hat{T}\) of \(T\) is described, such that there is an isomorphism between the geodesic billiard and a certain suspension of \(\hat{T}\). The correspondence under this isomorphism of invariant measures, Mather minimal measures, and of homological/cohomological data is described in detail.