×

Generalized radix representations and dynamical systems. I. (English) Zbl 1110.11003

The purpose of this paper is to study the dynamical system \(\tau_{{\mathbf r}}: \mathbb Z^d \to \mathbb Z^d\) (where \({\mathbf r} = (r_1,\dots,r_d)\in \mathbb R^d\)) that is defined by \(\tau_{{\mathbf r}}(a_1,\dots,a_d) =(a_2,\dots,a_d, - \lfloor r_1a_1 + \cdots + r_da_d \rfloor)\). For example, the authors consider the spectral radius of a related matrix, convexity properties, properties of ultimately periodic systems, long periods, and critical points.
The major reason for studying \(\tau_{{\mathbf r}}\) ist that these kinds of dynamical systems are intimately related to the so-called property (F) for Pisot numbers \(\beta>1\) (i.e. \(\mathbb Z[1/\beta]\cap [0,\infty)\) is exactly the set of positive real numbers with finite greedy \(\beta\)-expansion). Namely, if \(\beta> 1\) is an algebraic integer with minimal polynomial \((X-\beta)(X^{d-1} + r_2 X^{d-2} + \cdots + r_d)\) then \(\beta\) has property (F) if and only if for all \({\mathbf a}\in \mathbb Z^{d-1}\) there is \(k>0\) with \(\tau_{r_d,\dots,r_2}^k({\mathbf a}) =\mathbf{0}\). Further there is a similar relation to canonical number systems.

MSC:

11A67 Other number representations
37B10 Symbolic dynamics
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
PDFBibTeX XMLCite
Full Text: DOI Link