Numeration systems,fractals and stochastic processes

A numeration system Ω is a compactification of the set of real numbers keeping the actions of addition and positive multiplication in a natural way. That is, Ω is a compact metrizable space with #Ω≥2 to which ℝ acts additively andG acts multiplicatively satisfying the distributive law, whereG is a nontrivial closed multiplicative subgroup of ℝ₊. Moreover, the additive action is minimal and uniquely ergodic with 0-topological entropy, while the multiplication by λ has |log λ|-topological entropy attained uniquely by the unique invariant probability measure under the additive action. We construct Ω as above as a colored tiling space corresponding to a weighted substitution. This framework contains especially the substitution dynamical systems and β-transformation systems with periodic expansion of 1, both of which have discreteG. It also contains systems withG=ℝ₊. We study α-homogeneous cocycles on it with respect to the addition. They are interesting from the point of view of fractal functions or sets as well as self-similar processes. We obtain the zeta-functions of Ω with respect to the multiplication.