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Rank, covering number and a simple spectrum. (English) Zbl 0849.28008
Let $$T$$ be an ergodic automorphism on a standard Borel probability space $$(X, {\mathcal B}, \mu)$$. If $$m$$ and $$r$$ are positive integers $$(m\leq r)$$, the question of the existence of such $$T$$ having rank $$r$$ and maximal spectral multiplicity $$m$$ has been studied by several authors (see for example S. Ferenczi, J. Kwiatkowski and C. Mauduit [J. Anal. Math. 65, 45-75 (1995; Zbl 0833.28010)]). In fact, Kwiatkowski and Lacroix have recently announced that they have solved this problem completely. In this paper, an additional property is required of the transformation $$T$$. Roughly speaking, $$T$$ has covering number $$b$$ if there is a sequence of Rokhlin towers $$X_n$$ with $$\mu(X_n)\to b$$ as $$n\to \infty$$, and such that for every $$A\in {\mathcal B}$$, the set $$A\cap X_n$$ can be approximated arbitrarily closely by the levels of $$X_n$$. If $$b= 1$$, then $$T$$ has rank 1, and it always is true that if $$r= \text{rank}(T)$$, then $$r\cdot b\geq 1$$. The idea of positive covering number was introduced by King, Thouvenout and Ferenczi (using the term local rank one) [see S. Ferenczi, Ann. Inst. Henri Poincaré, Probab. Stat. 20, 35-51 (1984; Zbl 0535.28010)].
Let $$n$$ be an integer. The authors show that given $$b$$ and $$r$$ satisfying $${1\over n+ 1}\leq b< {1\over n}$$ and $$r\geq n+ 1$$, then there exists an ergodic $$T$$ with $$\text{rank}(T)= r$$ and covering number $$b$$. It is also shown that there exists $$T$$ having a simple spectrum $$(m= 1)$$, $$\text{rank}(T)= r$$ and covering number $$b$$ for any positive integer $$r\geq 2$$ and any real number $$0< b< 1$$ satisfying $$r\cdot b\geq 1$$. The examples used are Morse dynamical systems.

##### MSC:
 28D05 Measure-preserving transformations 47A35 Ergodic theory of linear operators
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