Generalised number systems and the dynamics beyond

Abstract:

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 digit. 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 important 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. Up to now they have not been studied very intensively, however, recent researches show up astonishing relations with generalised beta-transformations. It is planned to study several generalisations of asymmetric tent maps, for example, tent maps that consist of three linear parts, and to find and describe suitable geometric (fractal) representations.