Convergence in Möbius Number Systems

Abstract The Möbius number systems use sequences of Möbius transformations to represent the extended real line or, equivalently, the unit complex circle. An infinite sequence of Möbius transformations represents a point x on the circle if and only if the transformations, in the limit, take the uniform measure on the circle to the Dirac measure centered at the point x. We present new characterizations of this convergence. Moreover, we show how to improve a known result that guarantees the existence of Möbius number systems for some Möbius iterative systems. As Möbius number systems use subshifts instead of the whole symbolic space, we can ask what is the language complexity of these subshifts. We offer (under some assumptions) a sufficient and necessary condition for a number system to be sofic.