Transformations with highly nonhomogeneous spectrum of finite multiplicity
In an earlier paper [Invent. Math. 72, 299-314 (1983; Zbl 0519.28008)] the author constructed ergodic measure-preserving transformations T with arbitrary finite spectral multiplicities \(M_ T\). Using similar but more elaborate techniques, the author studies the set of all essential spectral multiplicities \({\mathcal M}_ T\) of T, the essential range of the multiplicity functions, so in particular \(M_ T=\sup {\mathcal M}_ T.\) \(D_ T=card {\mathcal M}_ T\) is called the degree of nonhomogenity of the spectrum of T. In 1966 Oseledec constructed an ergodic T with \(1<M_ T<\infty,\) but until 1983 most common examples were known to have \({\mathcal M}_ T=\{1\},\{\infty \}\quad or\quad \{1,\infty \}.\) At this time a number of new examples appeared, those of the author, the reviewer [Proc. Am. Math. Soc. 93, 303-306 (1985; Zbl 0551.28019)], \textit{J. Mathew} and \textit{M. G. Nadkarni} [Bull. Lond. Math. Soc. 16, 402-406 (1984; Zbl 0515.28010)] and \textit{M. Queffelec} [C. R. Acad. Sci., Paris, Sér. I 297, 317-320 (1983; Zbl 0537.28007)], but all of these had \({\mathcal M}_ T=\{1,k\}\) for some \(1<k<\infty.\) In this paper the author constructs examples with \(D_ T<\infty,\quad M_ T<\infty\) and \({\mathcal M}_ T\) satisfying \((i)\quad 1\in {\mathcal M}_ T,\) (ii) if \(m_ 1,m_ 2\in {\mathcal M}_ T\) then \(lcm(m_ 1,m_ 2)\in {\mathcal M}_ T.\) For example if p is prime \({\mathcal M}_ T=\{1,p-1,p(p- 1),...,p^ r(p-1)\}\) is a possibility which occurs. The basic technique used is the theory of approximation by period transformation.
ergodic measure-preserving transformations
scientific article; zbMATH DE number 3993906
scientific article