The analogue of overlap-freeness for the Fibonacci morphism

A 4−-power is a non-empty word of the form XXXX−, where X− is obtained from X by erasing the last letter. A binary word is called faux-bonacci if it contains no 4−-powers, and no factor 11. We show that faux-bonacci words bear the same relationship to the Fibonacci morphism that overlap-free words bear to the Thue-Morse morphism. We prove the analogue of Fife’s Theorem for faux-bonacci words, and characterize the lexicographically least and greatest infinite faux-bonacci words.