Substitution in Two Symbols and Transcendence
Kumiko Nishioka, Taka-aki Tanaka, Zhi-Ying WEN · Tokyo Journal of Mathematics · 1999
Let $A=\{a_{1}, \cdots, a_{n}\}$ be a finite nonempty set of symbols and let $A^{*}$ and $A^{\omega}$ denote the sets of all finite words over $A$ and all sequences $x_{0}x_{1}\cdots x_{k}\cdots(x_{k}\in A)$ , respectively.Let $\lambda$ be the empty word.A substitution (over $A$ ) is a map $\sigma:A\rightarrow A^{*}\backslash $ $\{\lambda\}$ , which has a natural extension to $\Omega=A^{*}\cup A^{\omega}$ by concatenation: $\sigma(x_{O}x_{1}\cdots)=$ $\sigma(x_{0})\sigma(x_{1})\cdots$ .If $a_{i}$ is a prefix of $\sigma(a_{i})$ and the length of $\sigma(a_{i})$ is greater than 1, then there is a unique $ w\in\Omega$ having a prefix $a_{i}$ and being a fixed point of $\sigma$ , which means that $\sigma(w)=w$ .Any real algebraic irrational $\theta$ can be uniquely expressed aswhere $m$ is a nonnegative integer depending on $\theta$ and $\epsilon_{k}=0$ or 1.The problem we are interested in is whether the sequence $\epsilon_{0}\epsilon_{1}\cdots\in\{0,1\}^{\omega}$ is a fixed point of any substitution over $\{0,1\}$ or not.Generally, for a fixed point $ w=x_{0}x_{1}\cdots$ of the given substitution $\sigma$ , we define the generating function of $w$ for $a_{i}$ by $f_{i}(z)=\sum_{k=0}^{\infty}\chi_{k}(w;a_{i})z^{k}$ , (where $\chi_{k}(w;a_{i})=1$ if $x_{k}=a_{i}$ , and otherwise $\chi_{k}(w;a_{i})=0$ , so that $\sum_{i=1}^{n}f_{i}(z)=\sum_{k=0}^{\infty}z^{k}=\frac{1}{1-z}$ .It is known that $f_{i}(z)(1\leq i\leq n)$ satisfy a Mahler type functional equation if $\sigma$ is of constant length, which means that each $\sigma(a_{i})(1\leq i\leq n)$ has the same length $\geq 2$ , and it is also known that if $\sigma$ is of nonconstant length, i.e., the lengths of $\sigma(a_{i})(1\leq i\leq n)$ are not equal, then we can construct $g_{1}(z),$ $\cdots,$ $g_{n}(z)\in Q [[z_{1}, \cdots, z_{n}]]$ satisfying a Mahler