Classification of all cycles of the parabolic map (Q1181148)

One parameter family of parabolic maps \(p_ \zeta(x)=\zeta x(2-x)\) is investigated. Determining all cycles of \(p_ \zeta (x)\) for all parameter values \(\zeta \) is the subject of the paper. To determine how the set of all fixed points of the nth iterate \(p^ n_ \zeta (x)\) is decomposed into cycles of \(p_ \zeta (x)\) the authors construct the inverse graph of \(p^ n_ \zeta(x)\) using \(2^ n\) functions of \(x\) that are inverse to the polynomial \(p^ n_ \zeta(x)\). These functions are labelled by finite sequences of integers and a total order relation on such sequences which corresponds to the order of the inverse functions is given. It is shown how the properties of these sequences give a complete description of the evolution of the graph in the parameter \(\zeta\) with respect to changes in its shape and in its bifurcation structure. These results enable the authors to determine how the set of fixed points of \(p^ n_ \zeta\) is decomposed into cycles of \(p_ \zeta\).