Rate of mixing for equilibrium states in negative curvature and trees

TL;DR

This paper surveys the construction of equilibrium states for geodesic flows in negatively curved spaces and trees, emphasizing their mixing rates, and introduces new examples of nonuniform tree lattices with exponential mixing properties.

Contribution

It provides a new construction of nonuniform tree lattices with exponentially mixing geodesic flows, covering arbitrary end spaces and growth types.

Findings

Exponential mixing achieved for various nonuniform tree quotients.

Construction of examples with arbitrary end spaces.

Demonstration of mixing properties in noncompact settings.

Abstract

In this survey based on the book by the authors [BPP], we recall the Patterson-Sullivan construction of equilibrium states for the geodesic flow on negatively curved orbifolds or tree quotients, and discuss their mixing properties, emphazising the rate of mixing for (not necessarily compact) tree quotients via coding by countable (not necessarily finite) topological shifts. We give a new construction of numerous nonuniform tree lattices such that the (discrete time) geodesic flow on the tree quotient is exponentially mixing with respect to the maximal entropy measure: we construct examples whose tree quotients have an arbitrary space of ends or an arbitrary (at most exponential) growth type.