The Analog of Overlap-Freeness for the Period-Doubling Sequence

Good words are defined to be binary words avoiding the factors 11 and 1001, and the patterns 0000 and 00010100. We show that good words bear the same relationship to the period-doubling sequence that overlap-free words bear to the Thue-Morse sequence. We prove an analog of Fife's theorem for good words, exhibit the lexicographically least and greatest infinite good words, and determine the patterns avoided by the period doubling word.