Envelope Words and Return Words Sequences in the Period-doubling Sequence

We consider the infinite one-sided sequence generated by the period-doubling substitution $\sigma(a,b)=(ab,aa)$, denoted by $\mathbb{D}$. Since $\mathbb{D}$ is uniformly recurrent, each factor $\omega$ appears infinite many times in the sequence, which is arranged as $\omega_p$ $(p\ge 1)$. Let $r_p(\omega)$ be the $p$-th return word over $\omega$. The main result is: for each factor $\omega$, the sequence $\{r_p(\omega)\}_{p\geq1}$ is $\Theta_1$ or $\Theta_2$, which are substitutive sequences and determined completely in this paper.