Hyperbolic groupoids: metric and measure

TL;DR

This paper develops new methods for constructing measures and metrics on hyperbolic groupoids, providing a novel approach to defining SRB measures on Smale spaces via Gromov hyperbolic graphs.

Contribution

It introduces a new way to define Patterson-Sullivan measures and metrics on hyperbolic groupoids, linking them to SRB measures on Smale spaces.

Findings

Constructed Patterson-Sullivan measure for hyperbolic groupoids

Defined a natural metric on the unit space of hyperbolic groupoids

Provided a new approach to SRB measures using Gromov hyperbolic graphs

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 using Gromov hyperbolic graphs.