Numeration systems as dynamical systems -- introduction

A numeration system originally implies a digitization of real numbers, but in this paper it rather implies a compactification of real numbers as a result of the digitization. By definition, a numeration system with G, where G is a nontrivial closed multiplicative subgroup of R+, is a nontrivial compact metrizable space Ω admitting a continuous (λω + t)-action of (λ, t) ∈ G × R to ω ∈ Ω, such that the (ω+ t)-action is strictly ergodic with the unique invariant probability measure μΩ, which is the unique G-invariant probability measure attaining the topological entropy | log λ| of the transformation ω 7→ λω for any λ 6= 1. We construct a class of numeration systems coming from weighted substitutions, which contains those coming from substitutions or β-expansions with algebraic β. It also contains those with G = R+. We obtained an exact formula for the ζ-function of the numeration systems coming from weighted substitutions and studied the properties. We found a lot of applications of the numeration systems to the β-expansions, Fractal geometry or the deterministic self-similar processes which are seen in [10]. This paper is based on [9] changing the way of presentation. The complete version of this paper is in [10]. 1. Numeration systems By a numeration system, we mean a compact metrizable space Ω with at least 2 elements as follows: (♯1) There exists a nontrivial closed multiplicative subgroup G of R+ and a continuous action λω + t of (λ, t) ∈ G × R to ω ∈ Ω such that λ(λω + t) + t = λλω + λt+ t. (♯2) The (ω + t)-action of t ∈ R to ω ∈ Ω is strictly ergodic with the unique invariant probability measure μΩ called the equilibrium measure on Ω. Consequently, it is invariant under the (λω + t)-action of (λ, t) ∈ G× R to ω ∈ Ω as well. (♯3) For any fixed λ0 ∈ G, the transformation ω 7→ λ0ω on Ω has the | logλ0|topological entropy. For any probability measure ν on Ω other than μΩ which is invariant under the λω-action of λ ∈ G to ω, and 1 6= λ0 ∈ G, it holds that hν(λ0) < hμΩ(λ0) = | logλ0|. The (ω + t)-action of t ∈ R to ω ∈ Ω is called the additive action or R-action, while the λω-action of λ ∈ G to ω ∈ Ω is called the multiplicative action or G-action. Note that if Ω is a numeration system, then Ω is a connected space with the continuum cardinality. Also, note that the multiplicative group G as above is either R+ or {λn; n ∈ Z} for some λ > 1. Moreover, the additive action is faithful, that is, ω + t = ω implies t = 0 for any ω ∈ Ω and t ∈ R. Matsuyama University, 790-8578 Japan, e-mail: [email protected] AMS 2000 subject classifications: primary 37B10.