Factors of characteristic words: Location and decompositions

Let α be an irrational number with 0<α<1, and let fα be the characteristic word of α. The location of a factor w of fα is defined to be the set of all positions in fα at which w occurs. In this paper, an explicit formula of the location of each factor w of fα is obtained. The decompositions REDα(w) and SODα(w) of fα 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. Melançon (1999), W.-T. Cao and Z.-Y. Wen (2003), I.M. Araújo and V. Bruyère (2005), and A. Glen (2006).