New spectral multiplicities for ergodic actions

Anton V. Solomko ¹

¹ Institute for Low Temperature Physics & Engineering National Academy of Sciences of Ukraine 47 Lenin Ave. Kharkov, 61164, Ukraine

Studia Mathematica, Tome 208 (2012) no. 3, pp. 229-247

Résumé

Let $G$ be a locally compact second countable Abelian group. Given a measure preserving action $T$ of $G$ on a standard probability space $(X, \mu)$, let $\mathcal M(T)$ denote the set of essential values of the spectral multiplicity function of the Koopman representation $U_T$ of $G$ defined in $L^2(X,\mu)\ominus \mathbb C$ by $U_T(g)f := f\circ T_{-g}$. If $G$ is either a discrete countable Abelian group or $\mathbb R^n$, $n\geq 1$, it is shown that the sets of the form $\{p,q,pq\}$, $\{p,q,r,pq,pr,qr,pqr\}$ etc. or any multiplicative (and additive) subsemigroup of $\mathbb N$ are realizable as $\mathcal M(T)$ for a weakly mixing $G$-action $T$.

Détail

DOI : 10.4064/sm208-3-3

Keywords: locally compact second countable abelian group given measure preserving action standard probability space mathcal denote set essential values spectral multiplicity function koopman representation defined ominus mathbb f circ g either discrete countable abelian group mathbb geq shown sets form pqr etc multiplicative additive subsemigroup mathbb realizable mathcal weakly mixing g action nbsp