# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] [Cha] R. V. Chacon,A geometric construction of measure preserving transformations, Proc. Fifth Berkeley Symposium of Mathematical Statistics and Probability, Vol. II, part 2, Univ. of California Press, 1965, pp. 335–360. [2] [Del] H. Delange,Generalisation du théorème de Ikehara, Ann. Sci. Ec. Norm. Sup.71, Fasc 3 (1954). [3] [delJ] A. del Junco,A transformation with simple spectrum which is not rank one, Canad. J. Math.29 (1977), 655–663. · Zbl 0343.28009 · doi:10.4153/CJM-1977-067-7 [4] [Fer1] S. Ferenczi,Systèmes localement de rang un, Ann. Inst. H. Poincaré Probab. Statist.20 (1984), 35–51. · Zbl 0535.28010 [5] [Fer2] S. Ferenczi,Tiling and local rank properties of the Morse sequence, Theoret. Comput. Sci., to appear. [6] [FeKw] S. Ferenczi and J. Kwiatkowski,Rank and spectral multiplicity, Studia Mathematica102 (2) (1992), 121–144. · Zbl 0809.28013 [7] [GoKwLiLe] G. R. Goodson, J. Kwiatkowski, M. Lemańczyk and P. Liardet,On the multiplicity function of ergodic group extensions of rotations, Studia Mathematica102 (2) (1992), 157–174. · Zbl 0830.28009 [8] [GoLe] G. R. Goodson and M. Lemańczyk,On the rank of a class of bijective substitutions, Studia Mathematica96 (1990), 219–230. · Zbl 0711.28007 [9] [Kea] M. Keane,Strongly mixing g-measures, Invent. Math.16 (1972), 309–353. · Zbl 0241.28014 · doi:10.1007/BF01425715 [10] [Kwi] J. Kwiatkowski,Isomorphism of regular Morse dynamical systems, Studia Mathematica72 (1982), 59–89. · Zbl 0525.28018 [11] [KwSi] J. Kwiatkowski and A. Sikorski,Spectral properties of G-symbolic Morse shifts, Bull. Soc. Math. France115 (1987), 19–33. · Zbl 0624.28014 [12] [Lem] M. Lemańczyk,Toeplitz Z 2-extensions, Ann. Inst. H. Poincaré Probab. Statist.24 (1988), 1–43. [13] [Mar] J. C. Martin,The structure of generalized Morse minimal sets on n symbols, Trans. Amer. Math. Soc.232 (1977), 343–355. · Zbl 0375.28010 [14] [Men1] M. Mentzen.Some examples of automorphisms with rank r and simple spectrum, Bull. Polish Acad. Sci. Math.35 (1987), 417–424. · Zbl 0675.28006 [15] [Men2] M. Mentzen,Thesis, preprint no 2/89, Nicholas Copernicus University, Toruń, 1989. [16] [Nar] W. Narkiewicz,Number Theory, World Scientific, Singapore, 1983. [17] [Par] W. Parry,Compact abelian group extensions of discrete dynamical systems, Z. Wahr. Verw. Gebiete13 (1969), 95–113. · Zbl 0184.26901 · doi:10.1007/BF00537014 [18] [Rob1] E. A. Robinson,Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math.72 (1983), 299–314. · Zbl 0519.28008 · doi:10.1007/BF01389325 [19] [Rob2] E. A. Robinson,Mixing and spectral multiplicity, Ergodic Theory and Dynamical Systems5 (1985), 617–624. · Zbl 0565.28013 · doi:10.1017/S0143385700003205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.