Some new cases of realization of spectral multiplicity function for ergodic transformations

Volume 206 / 2009

A. Katok, M. Lemańczyk Fundamenta Mathematicae 206 (2009), 185-215 MSC: 37A05, 28D05, 37A30, 47A10. DOI: 10.4064/fm206-0-11


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.


  • A. KatokDepartment of Mathematics
    The Pennsylvania State University
    University Park, PA 16802, U.S.A.
  • M. LemańczykFaculty of Mathematics and Computer Science
    Nicolaus Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland

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