On two conjectures of Shallit about Thue–Morse-like sequences

We study a class of infinite words $x_k$, $k\in N,k\ge 1$, recently introduced by J. Shallit. This class includes the Thue–Morse sequence $x_1$, the Fibonacci–Thue–Morse sequence $x_2$, and the Allouche–Johnson sequence $x_3$. Shallit stated and for $k=3$ proved two conjectures on properties of $x_k$. The first conjecture concerns the factor complexity, the second one the critical exponent of these words. We confirm the validity of both conjectures for every $k$.