On sequences resulting from iteration of modified quadratic and palindromic mappings

Blanchard and Fabre (1994) have considered description of symbolic systems by sequences obtained by iteration of two mappings g j and d j , defined by g j(u) = u[ u] j and d j = u[ u] R j , where the word [ u] j is obtained from a word u by removing the last j symbols. They stated a conjecture that the iteration of d j results in an automatic sequence. We prove here that d j is, while g j is not, a 2-automatic sequence. Our result is obtained by investigation of two wider classes of sequences.