作为动力系统的计算系统--导论
Numeration systems as dynamical systems -- introduction
摘要：
一个计数系统最初意味着实数的数字化，但在本文中，它更意味着实数的紧化作为数字化的结果。根据定义，具有$G$的计数系统，其中$G$是${\mathbb{R}}_+$的非平凡闭乘法子群，是非平凡紧致可度量化空间$Ω$，其允许$(λ,t)\in G\times{\mathbb{R}}$到$ω\inΩ$的连续$(λω+t)$ -作用，使得$(ω+t)$ -作用严格遍历唯一不变概率测度$μ_Ω$，它是唯一的$G$不变概率测度，对于任何$λ\ne 1$，它获得变换$ω\mapstoλω$的拓扑熵$|\logλ|$。我们构造了一类由加权代换产生的计数系统，它包含由代数$β$代换或$β$ -展开产生的计数系统。它也包含那些与$G={\mathbb{R}}_+$。本文给出了由加权代换产生的数制的$ζ$ -函数的一个精确公式，并研究了其性质。我们发现了大量的应用程序的计数系统的$β$ -展开，分形几何或确定性的自相似过程中看到的\cite{K4}。本文就是基于\cite{K3}改变呈现方式。本文的完整版本在\cite{K4}中。
英文摘要：
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 $Ω$ admitting a continuous $(λω+t)$-action of $(λ,t)\in G\times{\mathbb{R}}$ to $ω\inΩ$, such that the $(ω+t)$-action is strictly ergodic with the unique invariant probability measure $μ_Ω$, which is the unique $G$-invariant probability measure attaining the topological entropy $|\logλ|$ of the transformation $ω\mapstoλω$ for any $λ\ne 1$. 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 $G={\mathbb{R}}_+$. 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 \cite{K4}. This paper is based on \cite{K3} changing the way of presentation. The complete version of this paper is in \cite{K4}.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
