Spectral multiplicities for ergodic flows

TL;DR

This paper constructs weakly mixing ergodic flows with prescribed spectral multiplicities, demonstrating the flexibility of spectral properties in measure-preserving systems and extending results to certain group actions.

Contribution

It introduces a method to realize any subset of positive integers (containing 1 or 2) as the spectral multiplicity set of a weakly mixing flow, extending to some Abelian group actions.

Findings

Spectral multiplicity sets can be prescribed for ergodic flows.

Construction of flows with specific spectral properties.

Extension of results to actions of certain Abelian groups.

Abstract

Let $E$ be a subset of positive integers such that $E\cap\{1,2\}\ne\emptyset$. A weakly mixing finite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that the set of spectral multiplicities (of the corresponding Koopman unitary representation generated by $T$) is $E$. Moreover, for each non-zero $t\in\Bbb R$, the set of spectral multiplicities of the transformation $T_t$ is also $E$. These results are partly extended to actions of some other locally compact second countable Abelian groups.