zbMATH — the first resource for mathematics

On the multiplicity function of ergodic group extensions. II. (English) Zbl 0857.28012
It is an open problem in ergodic theory whether for an arbitrary set \(A\subseteq\mathbb{Z}^+\cup\{\infty\}\) there is an ergodic automorphism \(T:(X,{\mathcal B},\mu)\to(X,{\mathcal B},\mu)\) on a standard Borel probability space, whose set of essential values of the multiplicity function (of the induced unitary operator) is equal to \(A\). In this paper, this problem is almost completely solved with the proviso that \(1\in A\). It is still open for example, whether the multiplicity function can take a constant value (a.e.) different from 1. The authors build on earlier work concerning Morse automorphisms [see F. Blanchard and M. Lemańczyk, Topol. Methods Nonlinear Anal. 1, No. 2, 275-294 (1993; Zbl 0793.28012) and (Part I of the present note) G. R. Goodson, J. Kwiatkowski, M. Lemańczyk and P. Liardet, Stud. Math. 102, No. 2, 157-174 (1992; Zbl 0830.28009) for example, where perhaps it should be mentioned that the first named author is the son of the latter mentioned Kwiatkowski].
The authors show that \(T\) can be chosen to be weakly mixing, and if \(A\) is finite, \(T\) can be an analytic diffeomorphism on a finite-dimensional torus.

28D05 Measure-preserving transformations
47A35 Ergodic theory of linear operators
Full Text: DOI EuDML