Cyclically repetition-free words on small alphabets

Abstract

All sufficiently long binary words contain a square but there are infinite binary words having only the short squares 00, 11 and 0101. Recently it was shown by J. Currie that there exist cyclically square-free words in a ternary alphabet except for lengths 5, 7, 9, 10, 14, and 17. We consider binary words all conjugates of which contain only short squares. We show that the number c(n) of these binary words of length n grows unboundedly. In order for this, we show that there are morphisms that preserve circular square-free words in the ternary alphabet. (C) 2010 Published by Elsevier B.V.