Hyperbolic groupoids: metric and measure

Abstract

We construct Patterson–Sullivan measure and a natural metric on the unit space of a hyperbolic groupoid. In particular, this gives a new approach to defining SRB measures on Smale spaces and Anosov flows using Gromov hyperbolic graphs. Other results include a duality theory for hyperbolic groupoids graded by Hölder continuous Busemann cocycles, and a characterization of visual metrics.