Möbius number systems

Alexandr Kazda

Möbius number systems represent the extended real line or, equivalently, the unit complex circle by sequences of Möbius transformations. A Möbius number system consists of an iterative system of Möbius transformations and a subshift. In this paper we give an overview of the area of Möbius number systems. We are particularly interested in the conditions, under which a Möbius number system does or does not exists. We give an overview of known sufficient and necessary conditions on the iterative system and then introduce a necessary condition for the subshift. As Möbius number systems use subshifts instead of the whole symbolic space, we can ask what is the language complexity of these subshifts. We present a more user-friendly version of an already known condition for a number system to be sofic.