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 Kamae (Kamae, T. (2005), Numeration systems as dynamical systems, Preprint, available at http://www14.plala.or.jp/kamae).

This paper is based on Kamae, Numeration systems, fractals and stochastic processes, (to appear) changing the way of presentation. The complete version of this paper is in Kamae, Numeration systems as dynamical systems, Preprint, available at http://www14.plala.or.jp/kamae).