Binary patterns in the Prouhet-Thue-Morse sequence

We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all(finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence canactually be found in that sequence as segments (up to exchange of letters inthe infinite case). This result was previously attributed to unpublished workby D. Guaiana and may also be derived from publications of A. Shur onlyavailable in Russian. We also identify the (finitely many) finite binarypatterns that appear non trivially, in the sense that they are obtained byapplying an endomorphism that does not map the set of all segments of thesequence into itself