Automorphisms with finite exact uniform rank. (English) Zbl 0742.28007

Mentzen, Mieczysław K.

Stud. Math. 100, No. 1, 13-24 (1991).

Let \(X,{\mathcal B},\mu)\) be a Lebesgue probability space and \(T: X\to X\) an automorphism. Then \(T\) is said to have exact uniform rank at most \(r\) \((\hbox{EUR}(T)\leq r)\), if there exists a sequence of partitions \({\mathcal P}^ n\), \(n=1,2,\dots\) (of all of \(X\)), such that each \({\mathcal P}^ n\) consists of \(r\) pairwise disjoint stacks \((Q=(Q_ 0,\dots,Q_{n-1})\) is a stack of height \(n\) for \(T\) if \(Q_{i+1}=TQ_ i\), \(i=0,\dots,n-2)\), satisfying \({\mathcal P}^ n\to{\mathcal B}\) as \(n\to\infty\) and for each \(n\in\mathbb{N}\), the heights of all the stacks are equal. \(\hbox{EUR}(T)=r\) if \(r\) is the smallest number satisfying \(\hbox{EUR}(T)\leq r\).
The author shows that \(\hbox{EUR}(S)\leq\hbox{EUR}(T)\) whenever \(S\) is a factor of \(T\), but that \(\hbox{EUR}(T^ n)\leq| n|\hbox{EUR}(T)\) \((T^ n\) ergodic) fails to hold. These are results known to hold for other notions of rank [see for example D. S. Ornstein, D. J. Rudolph and B. Weiss, Mem. Am. Math. Soc. 262, 116 p. (1982; Zbl 0504.28019)]. The author is able to show that \(\hbox{EUR}(T^ n)\leq\hbox{EUR}(T)^{| n|}\) whenever \(T^ n\) is ergodic and \(\hbox{EUR}(T)<\infty\). Also it is shown that if \(\hbox{EUR}(T)<\infty\) then \(T\) is a finite extension of a transformation which has rational discrete spectrum.

Reviewer: G.R.Goodson (Towson)

MSC:

28D05

Measure-preserving transformations

Keywords:

ergodic automorphism; finite rank; exact uniform rank; stack; rational discrete spectrum