On spectral multiplicities of Gaussian actions

TL;DR

This paper explores the spectral multiplicities of Gaussian automorphisms and flows, demonstrating the diversity of spectral multiplicity sets, including mixing and non-mixing cases, and their variation within Gaussian flows.

Contribution

It constructs Gaussian automorphisms with prescribed spectral multiplicity sets and shows that spectral multiplicities can differ among automorphisms within a Gaussian flow.

Findings

Existence of Gaussian automorphisms with any prescribed spectral multiplicity set.

Spectral multiplicities can be different for automorphisms within the same Gaussian flow.

Some automorphisms are disjoint from all Gaussian actions.

Abstract

For any set $M$ of natural numbers there are mixing Gaussian automorphisms and non-mixing Gaussian automorphisms with singular spectrum (as well as some automorphisms which are disjoint from all Gaussian actions) such that $ M\cup\{\infty\}$ is the set of their spectral multiplicities. We show also that for a Gaussian flow $\{G_t\}$ the sets of spectral multiplicities for some automorphisms $G_t$, $t>0$, could be different.