Equidistribution, ergodicity and irreducibility in CAT(-1) spaces

TL;DR

This paper establishes an equidistribution theorem for boundary representations in CAT(-1) spaces, generalizing ergodic theorems and providing new insights into the irreducibility of boundary representations.

Contribution

It introduces a new equidistribution theorem for operator-valued measures in CAT(-1) spaces, extending classical ergodic results to a geometric group theory context.

Findings

Boundary representations are irreducible.

Established equidistribution of conformal densities.

Generalized Birkhoff's ergodic theorem to CAT(-1) spaces.

Abstract

We prove an equidistribution theorem a la Bader-Muchnik for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(-1) spaces thanks to an equidistribution theorem of T. Roblin. This result can be viewed as a generalization of Birkhoff's ergodic theorem for quasi invariant measures. In particular, this approach gives a dynamical proof of the fact that boundary representations are irreducible. Moreover, we prove some equidistribution results for conformal densities using elementary techniques from harmonic analysis.