We generalize an equidistribution theorem à 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.
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Adrien Boyer, Dustin Mayeda, Equidistribution, ergodicity and irreducibility associated with Gibbs measures. Comment. Math. Helv. 92 (2017), no. 2, pp. 349–387DOI 10.4171/CMH/414