Equidistribution, Ergodicity and Irreducibility in CAT(-1) spaces

We prove an equidistribution theorem à la Bader-Muchnik ([3]) 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 ([22]). This result can be viewed as a von Neumann’s mean 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. AMS subject classifications: Primary 37A25, 37A30; Secondary 43A65, 43A90. AMS keywords: Conformal densities, boundary representations, ergodic theorems, irreducibility, equidistribution. Adrien Boyer, Technion, Haifa, Israel. E-mail address: [email protected].