# Entropy and drift for Gibbs measures on geometrically finite manifolds

## TL;DR

This paper generalizes a key inequality relating entropy, drift, and critical exponent for Gibbs measures on geometrically finite manifolds, establishing conditions for measure equivalence and singularity.

## Contribution

It extends Guivarc'h's inequality to geometrically finite quotients of CAT(-1) spaces and characterizes when measures are equivalent or singular.

## Key findings

- Equality holds iff Gibbs density is equivalent to hitting measure.
- Hitting measure is singular to Gibbs density if the action is not convex cocompact.
- Provides conditions for measure equivalence in geometrically finite settings.

## Abstract

We prove a generalization of the fundamental inequality of Guivarc'h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of CAT(-1) metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is equivalent to the hitting measure. As a corollary, if the action is not convex cocompact, any hitting measure is singular to any Gibbs density.