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

We study a class of infinite words $\xx_k$, $k \in \N, k\geq 1$, recently introduced by J. Shallit. This class includes the Thue-Morse sequence $\xx_1$, the Fibonacci-Thue-Morse sequence $\xx_2$,and the Allouche-Johnson sequence $\xx_3$. Shallit stated and for $k=3$ proved two conjectures on properties of $\xx_k$. 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 $k$.