On the multiplicity function of generic group extensions with continuous spectrum (Q2716646)

Let \(T\) be an ergodic automorphism of a Lebesgue probability space. Denote by \(M(T)\) the set of essential spectral multiplicities of the corresponding unitary operator \(U_T: L^2(X,\mu)\to L^2(X,\mu)\), \(U_Tf(x)= f(Tx)\) (restricted to the orthogonal complement of the constant functions). Let \({\mathcal M}\subseteq \mathbb{Z}^+\cup \{\infty\}\) be an arbitrary subset. It is an open problem in ergodic theory whether there exists an ergodic automorphism \(T\) with \(M(T)={\mathcal M}\). A number of authors have considered this problem, the most general result so far being due to \textit{J. Kwiatkowski} and \textit{M. Lemańczyk} [Stud. Math. 116, 207--214 (1995; Zbl 0857.28012)]. They show that for any set \({\mathcal M}\subseteq \mathbb{Z}^+\cup \{\infty\}\) where \(1\in{\mathcal M}\), there is a weak mixing transformation with \(M(T)={\mathcal M}\). In this paper it is shown that for a generic set of automorphisms \(T\) (with respect to the usual topology), for any such \({\mathcal M}\) with \(1\in{\mathcal M}\), there exists a generic set of weakly mixing group extensions \(T'\) of \(T\) with \(M(T')={\mathcal M}\). NEWLINENEWLINENEWLINEWe should mention that examples with \(M(T)= \{2\}\) were previously constructed by the author [J. Dyn. Control Syst. 5, 149--152 (1999; Zbl 0943.37005)] and independently by \textit{V. V. Ryzhikov} [J. Dyn. Control Syst. 5, 145--148 (1999; Zbl 0954.37007)].