Factors of characteristic words: Location and decompositions

Let @a be an irrational number with 0<@a<1, and let f"@a be the characteristic word of @a. The location of a factor w of f"@a is defined to be the set of all positions in f"@a at which w occurs. In this paper, an explicit formula of the location of each factor w of f"@a is obtained. The decompositions RED"@a(w) and SOD"@a(w) of f"@a associated with w are established. The former one involves all return words of w while the latter one involves all occurrences of w and the corresponding separate factors and overlap factors. These results generalize the work by the present authors (2005), Z.-X. Wen and Z.-Y. Wen (1994), G. Melancon (1999), W.-T. Cao and Z.-Y. Wen (2003), I.M. Araujo and V. Bruyere (2005), and A. Glen (2006).

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