The analogue of overlap-freeness for the Fibonacci morphism

Abstract

A $4^{-}$-power is a non-empty word of the form $XXX{X}^{-}$, where $X^{-}$ is obtained from $X$ by erasing the last letter. A binary word is called {\em 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.