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Numeration Systems as Dynamical Systems: Introduction

Teturo Kamae
Lecture Notes-Monograph Series
Vol. 48, Dynamics & Stochastics (2006), pp. 198-211
Stable URL: http://www.jstor.org/stable/4356373
Page Count: 14
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Numeration Systems as Dynamical Systems: Introduction
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Abstract

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 $\mu _{\Omega}$, which is the unique G-invariant probability measure attaining the topological entropy |log λ| of the transformation ω → λω for any λ ≠ 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 $G={\Bbb R}_{+}$. 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].

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