On the rank of a class of bijective substitutions. (English) Zbl 0711.28007

Goodson, G. R.; Lemańczyk, M.

Stud. Math. 96, No. 3, 219-230 (1990).

It was conjectured by M. K. Mentzen that for any pair of rational numbers (k,n), where \(k\leq n\), it is possible to construct an ergodic automorphism with rank n and spectral multiplicity k. The case (1,n) was solved by M. K. Mentzen [Bull. Pol. Acad. Sci., Math. 35, 417-424 (1987; Zbl 0675.28006)]. In this paper it is shown that a general class of bijective substitutions over r symbols has rank r and this theorem is used to solve Mentzens problem for the case (2,n).

Reviewer: R.Cowen

MSC:

28D05

Measure-preserving transformations

Keywords:

Morse dynamical system; Morse substitution; ergodic automorphism; rank; spectral multiplicity; bijective substitutions