Genealogy of periodic points of maps of the interval

Abstract

Abstract. We describe the behavior of families of periodic points in one parameter families of maps of the interval which feature a transition from simple dynamics with finitely many periodic points to chaotic mappings. In particular, we give topological criteria for the appearance and disappearance of these families. Our results apply specifically to quadratic maps of the form F (x) = fuc(l -x).0. Introduction. In recent years, much attention has been paid to the dynamical properties of smooth mappings of the real line to itself. The logistic function 8p(x) = px(l -x) in particular has been studied by many authors. As the parameter p for this family is varied through positive values, the associated dynamical systems become increasingly more complex. For example, when 0 < p < I, there are only two fixed points. All other points tend either to one of these two fixed points or else to -oo under iteration. As p increases further, it can be shown that more and more periodic orbits appear in a regular fashion and that as long as p is less than approximately 3.57, the system remains relatively simple, i.e., there are only finitely many periodic points.On the other hand, when p > 4, the dynamical system is much more complicated yet still well understood. All points tend to -oo with the exception of a subset A of the unit interval which is mapped into itself. A is known to be homeomorphic to a Cantor set (see § 1 below) and, moreover, the restriction of gM to A is topologically conjugate to a one-sided shift automorphism on two symbols. In particular, there are 2" fixed points for gjj, so the dynamics here are quite a bit different from those for lower values of p. A major question in bifurcation theory is how does one progress from the simple dynamical systems for p < 3.57 to the more complex systems found for higher values of p. One of our goals in this paper is to investigate how the infinitely many periodic orbits arise as p approaches 4.More specifically, we study families of mappings f which resemble the logistic function in that/0 is identically 0 and/, has an invariant set A^ similar to the above when p > b. The precise hypotheses are given in § 1. For these mappings we show that a particular point arises via a finite sequence of simple bifurcations. We catalogue these bifurcations in the sense that we present an algorithm that decides which lower period points give rise via a sequence of bifurcations to a given