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arXiv · 2006.07238

Nonsingular Gaussian actions: beyond the mixing case

Amine Marrakchi·Stefaan Vaes
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Abstract

Every affine isometric action $α$ of a group $G$ on a real Hilbert space gives rise to a nonsingular action $\hatα$ of $G$ on the associated Gaussian probability space. In the recent paper [AIM19], several results on the ergodicity and Krieger type of these actions were established when the underlying orthogonal representation $π$ of $G$ is mixing. We develop new methods to prove ergodicity when $π$ is only weakly mixing. We determine the type of $\hatα$ in full generality. Using Cantor measures, we give examples of type III$_1$ ergodic Gaussian actions of $\mathbb{Z}$ whose underlying representation is non mixing, and even has a Dirichlet measure as spectral type. We also provide very general ergodicity results for Gaussian skew product actions.

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Amine Marrakchi, Stefaan Vaes. 2021-12-10. Nonsingular Gaussian actions: beyond the mixing case. https://doi.org/10.1016/j.aim.2022.108190

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