Some new cases of realization of spectral multiplicity function for ergodic transformations
Volume 206 / 2009
Fundamenta Mathematicae 206 (2009), 185-215
MSC: 37A05, 28D05, 37A30, 47A10.
DOI: 10.4064/fm206-0-11
Abstract
Given a countable Abelian group $\mathbb G$, its automorphism $w$ for which $w^M={\rm Id}$, and a subgroup ${\mathbb F}\subset \mathbb G$ we define $$M({\mathbb G},w,{\mathbb F})=\{\sharp(\{w^i\chi:i \in{\mathbb Z}\}\cap {\mathbb F}): \chi\in {\mathbb F}\setminus\{0\}\}.$$ We prove that each finite set of the form $M({\mathbb G},w,{\mathbb F})\cup\{2\}$ is realized as the set of essential values of the multiplicity function of the Koopman operator of some weakly mixing automorphism.
