Power-free complementary binary morphisms

We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary morphisms. Such morphisms are defined over a 2-letter alphabet, and map the letters 0 and 1 to complementary words. We prove that every prefix of the famous Thue–Morse word t gives a complementary morphism that is $3^{+}$-free and hence α-free for every real number $\alpha >3$. We also describe, using a 4-state binary finite automaton, the lengths of all prefixes of t that give cubefree complementary morphisms. Next, we show that 3-free (cubefree) complementary morphisms of length k exist for all $k\notin \{3,6\}$. Moreover, if k is not of the form $3\cdot 2^n$, then the images of letters can be chosen to be factors of t. Finally, we observe that each cubefree complementary morphism is also α-free for some $\alpha <3$; in contrast, no binary morphism that maps each letter to a word of length 3 (resp., a word of length 6) is α-free for any $\alpha <3$.

In addition to more traditional techniques of combinatorics on words, we also rely on the Walnut theorem-prover. Its use and limitations are discussed.