Hyperbolic geometry and pointwise ergodic theorems

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.

Editorial changes only, some references added and typos corrected. No change in the main results and their proofs