Groups, Geometry, and Dynamics

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Volume 11, Issue 3, 2017, pp. 777–818
DOI: 10.4171/GGD/415

Published online: 2017-08-22

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

Adrien Boyer[1]

(1) Technion – Israel Institute of Technology, Haifa, Israel

We prove an equidistribution theorem à 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 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.

Keywords: Conformal densities, boundary representations, ergodic theorems, irreducibility, equidistribution

Boyer Adrien: Equidistribution, ergodicity and irreducibility in CAT(–1) spaces. Groups Geom. Dyn. 11 (2017), 777-818. doi: 10.4171/GGD/415