Adjacency and Parity Relations of Words in Discrete Dynamical Systems

Words on two letters, or their equivalent representation by α sequences, label the branches of the inverse graph of the nth iterate of the parabolic map pζ(x)=ζx(2−x) of the real line. The abstract properties of words control the evolution of this graph in the content parameter ζ. In particular, properties of words (α sequences) control the process of creation and bifurcation of fixed points. The subset of lexical words of length n−1 or the corresponding set of lexical α sequences of degree D=n−1 are key entities in this description, as are the divisor set of lexical words of degree D such that 1+D divides n. The parity, even or odd, of the length of the lexical sequences in the divisor set controls the motion, from left to right or right to left, of the central point (1, x(ζ)) of the inverse graph through the midpoint (1, 1), as the content parameter ζ increase. It is proved in this paper that adjacent sequences in the ordered divisor set alternate in the parity of their lengths, this then corresponding to an oscillatory motion of the central point back and forth through the central point. The abstract parity property of words thus corresponds to an important property of the inverse graph in its evolution in the content parameter.