Sex: Dynamics, topology, and bifurcations of complex exponentials

Abstract

Our goal in this paper is to describe some of the interesting topology that arises in the dynamics of entire functions such as the complex exponential family Eλ(z)= λe. We will see that the important invariant sets for this family possesses a extremely rich topological structure, including such objects as Cantor bouquets, Knaster continua, hair transplants, and explosion points. For a complex analytic function E, the interesting orbits lie in the Julia set, which we denote by J (E). This is the set on which the map is chaotic. For the exponential family, the Julia set of Eλ has three characterizations: (1) J (Eλ) is the set of points at which the family of iterates of Eλ, {En λ} is not a normal family in the sense of Montel. This is the characterization that is most useful to prove theorems. (2) J (Eλ) is the closure of the set of repelling periodic points of Eλ. This is the dynamical definition of the Julia set. (3) J (Eλ) is the closure of the set of points whose orbits tend to ∞. This is the characterization that is most useful to compute the Julia set. We remark that characterization 3 differs from the case of polynomial iterations, where the Julia set is the boundary of the set of escaping orbits. The reason for the difference is Eλ has an essential singularity at ∞, while polynomials have superattracting fixed points at ∞. The equivalence of (1) and (2) was shown by Baker, see [6]. The equivalence of (1) and (3) is shown in [19]. In this paper we will concentrate on the dynamics of Eλ where λ is real. For λ positive, the Julia set for Eλ undergoes a remarkable transformation as λ passes through 1/e. We will show below that Eλ possesses an attracting fixed point when 0< λ< 1/e. All points in the left half plane have orbits that tend to this fixed point. Indeed, the full basin of attraction of this fixed point is open and dense in the plane.