Generalised number systems and the dynamics beyond

Beta-expansions are generalised positional notations with respect to a base that is not necessarily an integer. In the present project we are mainly interested in the induced digit sequences.over the (finite) set of digits. They form a symbolic dynamical system. In the classical case (greedy expansion) these so-called beta-shifts have been studied in enumerous research papers. It is well known that in the case of a sofic shift there exist imprtant connections with substitution dynamical systems. Beta-exopansions can be generalised in many different ways. For example, one may allow negative digits, too. As keyword we want to mention symmetric beta-expansions here. This topic is of growing interest, however, there are no known relations between the induced generalised beta-shifts and substitution dynamical systems. The principle goal of the present project is to establish these connections. We want to characterise them and to study the consequences in several directions. A second aim concerns tent maps. A Tent map is a continuous composite of two linear functions that acts on the unit interval. Recent researches show up astonishing relations with generalised beta- expansions. It is planned to consider several generalisations of asymmetric tent maps, for example, tent maps that consist of three linear parts, and to define and study suitable geometric (fractal) representations.

The project dealt with different notions of generalised number systems, that is the representation of numbers and sets of numbers. Such investigations are always intimately connected with dynamical systems. At first we want to mention the representation of integers. Here we generalise the Dumont-Thomas numeration. It is based on substitutive dynamical systems and is related with representations with respect to a linear recurrent sequence. Our generalisation considers negative integers, too, that do not appear in the classical Dumont-Thomas numeration. In this context we are also interested in representations with respect to alternating recurrent sequences as, for example, Knuth's negaFibonacci expansion. It is also possible to represent (positive) real numbers with respect to a substitution. This real version of the Dumont-Thomas numeration provides the connection between substitution dynamical systems and the dynamics induced by beta-expansions. In another article we generalise this real Dumont-Thomas numeration and show that substitutions are also closely related with generalised beta-expansions. In a further paper we deal with representations of complex numbers. Here we present and investigate the zeta-expansion as a complex version of the (real) beta-expansion. It turns out that many properties of the beta-expansion can be transferred to the zeta-expansion in some way. Surprisingly, the zeta-expansion is related with shift radix systems, too. Another issue of the project concerned geometric representations of dynamical system. They often have a fractal shape and are also interesting in a popular-scientific way. In particular, on one side we study Rauzy fractals that arise from substitutive dynamical systems. Here we investigate the (set-theoretical) intersection of Rauzy fractals of substitutions with the same incidence matrix. On the other side we are interested in fractals of so-called tent-maps, that are continuous composites of two linear functions that act on the unit interval.