Ergodic theory of affine isometric actions on Hilbert spaces
Abstract
The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a oneparameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the PattersonSullivan theory as well as LyonsPemantle work on treeindexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type $\mathrm{III}_1$.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.04272
 Bibcode:
 2019arXiv191104272A
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Group Theory;
 Mathematics  Operator Algebras;
 Mathematics  Probability
 EPrint:
 62 pages