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arXiv · 1911.04272

Ergodic theory of affine isometric actions on Hilbert spaces

Yuki Arano·Yusuke Isono·Amine Marrakchi
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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 one-parameter 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 Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed 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 $\mathrm{III}_1$.

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Yuki Arano, Yusuke Isono, Amine Marrakchi. 2019-11-11. Ergodic theory of affine isometric actions on Hilbert spaces. https://arxiv.org/abs/1911.04272

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