Numeration systems as dynamical systems -- introduction

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 ${\mathbb{R}}_+$, is a nontrivial compact metrizable space $\Omega$ admitting a continuous $(\lambda\omega+t)$-action of $(\lambda,t)\in G\times{\mathbb{R}}$ to $\omega\in\Omega$, such that the $(\omega+t)$-action is strictly ergodic with the unique invariant probability measure $\mu_{\Omega}$, which is the unique $G$-invariant probability measure attaining the topological entropy $|\log\lambda|$ of the transformation $\omega\mapsto\lambda\omega$ for any $\lambda\ne 1$. We construct a class of numeration systems coming from weighted substitutions, which contains those coming from substitutions or $\beta$-expansions with algebraic $\beta$. It also contains those with $G={\mathbb{R}}_+$. We obtained an exact formula for the $\zeta$-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 $\beta$-expansions, Fractal geometry or the deterministic self-similar processes which are seen in \cite{K4}. This paper is based on \cite{K3} changing the way of presentation. The complete version of this paper is in \cite{K4}.