Continuity of the drift in groups acting on strongly hyperbolic spaces

TL;DR

This paper extends the Avalanche Principle to hyperbolic spaces, providing a geometric framework to study the continuity of drift and limit points for Markov processes in strongly hyperbolic spaces.

Contribution

It introduces a geometric version of the Avalanche Principle applicable to hyperbolic spaces, enabling new continuity results for drift and limit points of stochastic processes.

Findings

Continuity of drift for Markov processes in hyperbolic spaces

Extension of Avalanche Principle to geometric hyperbolic setting

Continuity of the limit point of the process

Abstract

The Avalanche principle, in its original setting, together with large deviations yields a systematic way of proving the continuity of the Lyapunov exponent. In this text we present a geometric version of the Avalanche Principle in the context of hyperbolic spaces, which will extend the usage of these techniques to study the drift in such spaces. This continuity criteria applies not only to the drift but also to the limit point of the process itself. We apply this abstract result to derive continuity of the drift for Markov processes in strongly hyperbolic spaces.