Equidistribution, ergodicity and irreducibility associated with Gibbs measures

TL;DR

This paper extends equidistribution and ergodic theorems to operator-valued measures derived from Gibbs measures in negatively curved spaces, establishing irreducibility of associated boundary representations.

Contribution

It generalizes equidistribution results to operator-valued measures linked with Gibbs measures and proves irreducibility of boundary representations in this context.

Findings

Established a generalized equidistribution theorem for operator-valued measures.

Proved irreducibility of boundary representations associated with Gibbs measures.

Combined functional analytic and dynamical tools for new theoretical insights.

Abstract

We generalize an equidistribution theorem \`a la Bader-Muchnik for operator-valued measures constructed from a family of boundary representations associated with Gibbs measures in the context of convex cocompact discrete group of isometries of a simply connected connected Riemannian manifold with pinched negative curvature. We combine a functional analytic tool, namely the property RD of hyperbolic groups, together with a dynamical tool: an equidistribution theorem of Paulin, Pollicott and Schapira inspired by a result of Roblin. In particular, we deduce irreducibility of these new classes of boundary representations.