Classification of all cycles of the parabolic map

Abstract

The problem of determining all cycles of the parabolic map p ζ( x) for all parameters values ζ ϵ (0,∞) and all xϵ (0,∞) is addressed. The method is one of decomposing the set of all fixed points of the nth iterate p ζ n ( x), for arbitrary n, into cycles of p ζ( x). Toward this goal, the inverse graph of p ζ n ( x) is constructed for all ζ and its complete description given in terms of sequences α=( α 0, α 1, α 2,…), each α i a nonnegative integer, and their negatives, and a total order relation on all such sequences. It is shown how the properties of these sequences give a complete description of the evolution of the graph in the parameter ζ with respect to changes in its shape and in its bifurcation structure. These results are then used to infer properties of the decomposition of the set of fixed points of p ζ n into cycles of p ζ.