Hyperbolic geometry and pointwise ergodic theorems

TL;DR

This paper proves pointwise ergodic theorems for natural averages on rank-one simple Lie groups using geometric methods in hyperbolic space, extending previous radial case results without relying on spectral theory.

Contribution

It introduces a novel geometric approach to pointwise ergodic theorems on hyperbolic spaces, surpassing prior spectral theory-based methods.

Findings

Established pointwise ergodic theorems for a broad class of averages on rank-one Lie groups

Extended results beyond the radial case previously studied

Developed a new geometric proof technique independent of spectral theory

Abstract

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.