## Dynamics, Geometry, & Groups Seminar

### Friday, November 1st, 2019

**Time:** 10:30 a.m** Place:** Jeffery Hall 319

**Speaker:** Giulio Tiozzo (Queen's University)

**Title:** Entropy and drift for Gibbs measures on geometrically finite manifolds.

**Abstract:** The boundary of a simply connected, negatively curved manifold carries two natural types of measures: on one hand, Gibbs measures such as the Patterson-Sullivan measure and the SRB measure. On the other hand, harmonic measures arising from random walks. We prove that the absolute continuity between a harmonic measure and a Gibbs measure is equivalent to a relation between entropy, drift and critical exponent, extending the previous formulas of Guivarc’h, Ledrappier, and Blachere-Haissinsky-Mathieu. This shows that if the manifold (or more generally, a CAT(-1) space) is geometrically finite but not convex cocompact, harmonic measures are singular with respect to Gibbs measures.