A survey on spectral multiplicities of ergodic actions

TL;DR

This survey explores which spectral multiplicity sets can be realized by various classes of ergodic transformations, detailing constructions and generalizations to group actions.

Contribution

It provides a comprehensive overview of realizable spectral multiplicity sets for different ergodic actions and discusses their constructions and generalizations.

Findings

Characterization of spectral multiplicity sets for ergodic transformations

Explicit constructions for various spectral multiplicity sets

Extensions to actions of Abelian locally compact groups

Abstract

Given a transformation $T$ of a standard measure space $(X,\mu)$, let $\Cal M(T)$ denote the set of spectral multiplicities of the Koopman operator $U_T$ defined in $L^2(X,\mu)\ominus\Bbb C$ by $U_Tf:=f\circ T$. It is discussed in this survey paper which subsets of $\Bbb N\cup\{\infty\}$ are realizable as $\Cal M(T)$ for various $T$: ergodic, weakly mixing, mixing, Gaussian, Poisson, ergodic infinite measure preserving, etc. The corresponding constructions are considered in detail. Generalizations to actions of Abelian locally compact second countable groups are also discussed.