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We consider the infinite one-sided sequence over alphabet $\{a,b\}$ generatedby the period-doubling substitution $\sigma(a)=ab$ and $\sigma(b)=aa$, denotedby $\mathbb{D}$. 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 $\{r_p(\omega)\}_{p\geq1}$ is$\Theta_1$ or $\Theta_2$. Both of them are substitutive sequences anddetermined completely in this paper. (2) For any factor $\omega$ in $\Theta_1$(resp. $\Theta_2$), the return word sequence $\{r_p(\omega)\}_{p\geq1}$ isstill $\Theta_1$ or $\Theta_2$. We call it the reflexivity property of thereturn word sequence. As an application, we introduce a notion of spectrum forstudying some typical combinatorial properties, such as separated, adjacent andoverlapped.

Envelope Words and the Reflexivity of the Return Word Sequences in the Period-doubling Sequence