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A density theorem for (multiplicity, rank) pairs. (English) Zbl 0833.28010
Recall that associated with a dynamical system $$(X,T, \mu)$$ there are certain invariants. The rank $$r$$, a conjugacy invariant, is the number of Rokhlin towers necessary to generate the system. The (spectral) multiplicity $$m$$, a spectral invariant, is the number of cyclic subspaces necessary to span the $$L^2$$-space for the associated $$L^2$$- operator. It is known that for a dynamical system $$m \leq r$$ (Chacon). M. K. Mentzen [Bull. Pol. Acad. Sci., Math. 35, 417-424 (1987; Zbl 0675.28006)] conjectured that for each pair $$(m,r)$$ with $$m \leq r$$ (natural numbers), it should be possible to construct a dynamical system with multiplicity $$m$$ and rank $$r$$, and he showed this is possible for pairs of the form $$(1,n)$$. Subsequently, this reviewer and M. Lemańczyk [Stud. Math. 96, No. 3, 219-230 (1990; Zbl 0711.28007)] showed that $$(2,n)$$ can be achieved. More recently, other authors have constructed pairs of the form $$(n,n)$$, $$(n,2n)$$ and $$(p,p - 1)$$, $$p$$ prime.
In this paper the authors construct a new class of examples giving rise to pairs of the form $$(d,n)$$ for all $$n \geq 3$$ and all $$d$$ dividing an arithmetic function $$\psi (n)$$. In particular, it follows that for fixed $$d$$, the set of possible $$n$$ is of density one in $${\mathbf N}$$.

##### MSC:
 28D10 One-parameter continuous families of measure-preserving transformations
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##### References:
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