Envelope Words and the Reflexivity of the Return Word Sequences in the Period-doubling Sequence
We consider the infinite one-sided sequence over alphabet $\{a,b\}$ generated by the period-doubling substitution $σ(a)=ab$ and $σ(b)=aa$, denoted by $\mathbb{D}$. Let $r_p(ω)$ be the $p$-th return word of factor $ω$. The main result of this paper is twofold. (1) For any factor $ω$ in $\mathbb{D}$, the return word sequence $\{r_p(ω)\}_{p\geq1}$ is $Θ_1$ or $Θ_2$. Both of them are substitutive sequences and determined completely in this paper. (2) For any factor $ω$ in $Θ_1$ (resp. $Θ_2$), the return word sequence $\{r_p(ω)\}_{p\geq1}$ is still $Θ_1$ or $Θ_2$. We call it the reflexivity property of the return word sequence. As an application, we introduce a notion of spectrum for studying some typical combinatorial properties, such as separated, adjacent and overlapped.