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• W1554324594 abstract "<!-- *** Custom HTML *** --> 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 $(lambdaomega+t)$-action of $(lambda,t)in Gtimes{mathbb R}$ to $omegainOmega$, 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 $|loglambda|$ of the transformation $omegamapstolambdaomega$ for any $lambdane 1$. 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), <i>Numeration systems as dynamical systems</i>, Preprint, available at http://www14.plala.or.jp/kamae). This paper is based on Kamae, <i>Numeration systems, fractals and stochastic processes</i>, (to appear) changing the way of presentation. The complete version of this paper is in Kamae, <i>Numeration systems as dynamical systems</i>, Preprint, available at http://www14.plala.or.jp/kamae)." @default.
• W1554324594 created "2016-06-24" @default.
• W1554324594 creator A5019227055 @default.
• W1554324594 date "2006-01-01" @default.
• W1554324594 modified "2023-09-27" @default.
• W1554324594 title "Numeration systems as dynamical systems–introduction" @default.
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• W1554324594 doi "https://doi.org/10.1214/074921706000000220" @default.
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