A Symmetric Random Walk defined by the Time-One Map of a Geodesic Flow

TL;DR

This paper investigates a symmetric random walk driven by the time-one map of a geodesic flow on hyperbolic manifolds, establishing conditions for stationary measures and exploring related dynamical properties.

Contribution

It provides necessary and sufficient conditions for the existence of volume-equivalent stationary measures for such random walks.

Findings

Conditions for stationary measures equivalent to volume are characterized.

Dynamical consequences of these measures are analyzed.

The work links geodesic flow dynamics with random walk behavior.

Abstract

In this note we consider a symmetric random walk defined by a $(f,f^{-1})$ Kalikow type system, where $f$ is the time-one map of the geodesic flow corresponding to an hyperbolic manifold. We provide necessary and sufficient conditions for the existence of an stationary measure for the walk that is equivalent to the volume in the corresponding unit tangent bundle. Some dynamical consequences for the random walk are deduced in these cases.