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 $\sigma (a)=ab$ and $\sigma (b)=aa$, denoted by $\mathbb{D}$. Let $r_p(\omega )$ be the $p$-th return word of factor $\omega$. The main result of this paper is twofold. (1) For any factor $\omega$ in $\mathbb{D}$, the return word sequence $\{r_p(\omega ){\}}_{p\ge 1}$ is ${\mathrm{\Theta }}_1$ or ${\mathrm{\Theta }}_2$. Both of them are substitutive sequences and determined completely in this paper. (2) For any factor $\omega$ in ${\mathrm{\Theta }}_1$ (resp. ${\mathrm{\Theta }}_2$), the return word sequence $\{r_p(\omega ){\}}_{p\ge 1}$ is still ${\mathrm{\Theta }}_1$ or ${\mathrm{\Theta }}_2$. 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.