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Abstract
A real Möbius iterative system is an action of a free semigroup of finite words acting via Möbius transformations on the extended real line. Its convergence space consists of all infinite words, such that the images of the Cauchy measure by the prefixes of the word converge to a point measure. A Möbius number system consists of a Möbius iterative system and a subshift included in the convergence space, such that any point measure can be obtained as the limit of some word of the subshift. We give some sufficient conditions on sofic subshifts to form Möbius number systems. We apply our theory to several number systems based on continued fractions.
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