On ergodic transformations with homogeneous spectrum
An old question of Rokhlin asks whether there exists an ergodic automorphism having a homogeneous spectrum of multiplicity two (i.e., the multiplicity function of the induced unitary operator takes the unique value 2, almost everywhere with respect to the maximal spectral type). It was recently shown that if \(A\) is any subset of \(\mathbb{Z}^+\cup \{\infty\}\) with \(1\in A\), then there is an ergodic automorphism whose multiplicity function has range \(A\) [see \textit{J. Kwiatkowski} and \textit{M. Lemańczyk}, Stud. Math. 116, 207-215 (1995; Zbl 0857.28012)], but the more general question is still open (for example, is there an ergodic automorphism having \(A= \{n\}\), \(n>2 \)?) The question of Rokhlin is answered affirmatively, as the author shows that for a generic set of ergodic transformations \(T\), \(T\times T\) has a homogeneous spectrum of multiplicity two. It should be noted that \textit{V. V. Ryzhikov} has simultaneously and independently solved this problem in a similar way [J. Dyn. Control Syst. 5, 145-148 (1999)]. See also the survey article of the reviewer [J. Dyn. Control Syst. 5, 173-226 (1999)], which gives a detailed look at the solution of Ryzhikov and also surveys recent results concerning the spectral theory of ergodic dynamical systems. The author is also able to answer affirmatively a conjecture of A. Katok concerning the range of the multiplicity function of Cartesian powers \(T\times T\times T\) (and higher order powers), for the generic transformation \(T\).
homogeneous spectrum of multiplicity two
range of the multiplicity function of Cartesian powers
scientific article; zbMATH DE number 1431877