Ergodicity of unipotent flows and Kleinian groups

TL;DR

This paper proves that a specific unipotent flow on the frame bundle of certain hyperbolic 3-manifolds is ergodic under conditions related to the critical exponent, advancing understanding of dynamical systems in hyperbolic geometry.

Contribution

It establishes ergodicity of one-dimensional unipotent flows on the frame bundle of convex cocompact hyperbolic 3-manifolds when the critical exponent exceeds one, linking geometric properties to dynamical behavior.

Findings

Ergodicity of unipotent flows for delta > 1

Connection between critical exponent and flow dynamics

Advancement in hyperbolic geometry and dynamical systems

Abstract

Let M be a non-elementary convex cocompact hyperbolic 3 manifold and delta the critical exponent of its fundamental group. We prove that a one-dimensional unipotent flow for the frame bundle of M is ergodic for the Burger-Roblin measure provided that delta>1.