Dynamical systems and digital representations in nonarchimedean fields

The representation of a number in base ten is the best known example of a digital expansion. This concept was generalized in several directions. Both expansions where the set of numbers to be represented is generalized and expansions where the base is generalized were considered. Many of these expansions are generated by dynamical systems. It turns out, that the underlying algorithms can be formulated not only in archimidean fields like the real or complex numbers, but also in non archimedean fields, like the formal Laurent series, or the p-adic numbers. As the properties of most of these Algorithms have been analyzed in the archimidean case, the corresponding theory is rather incomplete for the non archimedean counterparts. In the present project, we study metric properties of number systems for Laurentseries and p-adic numbers. In particular, we consider the analogues of canonical number systems and beta-expansions. Furthermore, we consider the relation of these expansions to fractal tilings.

The first paper is devoted to a new notion of continued fractions in the field of Laurent series over a finite field. The definition of this kind of continued fraction algorithm is based on so called beta-expansions. We prove some ergodic properties and compute the Hausdorrf dimensions of bounded type continued fraction sets. The second paper provides a common notation of several known digit systems in number fields as well as function fields. We consider digit representations in the polynomial rings over an euclidian ring. We derive a general concept that generalises both canonical number systems and digit systems over finite fields. Due to the fact that we allow quite general sets of digits, several new phenomena occur in this context. The third paper is still under preparation. In this paper, we generalise the concept of digit systems defined in the second paper to base polynomials in more than one variable. Although the characterization problem becomes more difficult, we are able to establish connections to digit systems in one variable and prove nontrivial results. We prove that the effect of addition of one on the digit array can be modelled by means of cellular automata. Furthermore, we provide an algorithmic construction for these "cellular odometers" for a given base polynomial.