See Halava, Vesa, Harju, Tero, Karhumäki, Juhani
Decidability of the binary infinite Post correspondence problem. If the alphabet consists of $\le 2$ letters, then the problem is decidable, if the number of letters is at least 7, then the problem is undecidable. The latter result is proved in Y. Matiyasevich, G. Sénizergues, Decision problems for semi-Thue systems with a few rules, in: Proceedings, 11th Annual IEEE Symposium on Logic in Computer Science, New Brunswick, NJ, 27–30 July 1996, IEEE Computer Society, Silver Spring, MD, pp. 523–531.