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Automorphisms with finite exact uniform rank. (English) Zbl 0742.28007
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.
##### MSC:
 28D05 Measure-preserving transformations
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