Equidistribution, ergodicity and irreducibility in CAT(-1) spaces
  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. 
Adrien Boyer
math.GR math.DS math.RT