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.


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