Using external rays to classify parameter space for non-autonomous polynomial dynamical systems
In this work I consider the dynamics arising from the iteration of an arbitrary sequence of polynomials with uniformly bounded degrees and coefficients and show that, as parameters vary within a single hyperbolic component in parameter space, certain properties of the corresponding Julia sets are preserved. In particular, I show that if the sequence is hyperbolic and all the Julia sets are connected, then the whole basin of attraction of infinity moves holomorphically. This extends also to the landing points of external rays and the resultant holomorphic motion of the Julia sets coincides with that obtained earlier by Comerford using grand orbits. In addition, combinatorial rigidity is established in the sense that if a finite set of external rays separate the Julia set for a particular parameter value, then the rays with the same external angles separate the Julia set for every parameter in the same hyperbolic component. In the case that the degrees of the polynomials in the sequence are all the same, I show that the only possible combinatorial configurations of the rays are essentially those that arise classically, but also that the assumption of constant degree is necessary here. ^ I also look at the converse problem; if one begins with the assumption that a sequence of rays with prescribed, combinatorially valid angles lands at a point in the Julia set for some parameter, then it is possible to begin to analyze the set of paramters for which this sequence of rays lands also lands at a single point. To this end, I define a generalized orbit portrait analagous to that defined by Douady and Hubbard and appropriate to the non-autonomous context. I then define a generalized parameter wake as the maximal set in parameter space such that membership to this set guarantees that the polynomial sequence acquired from the member parameter possesses a given orbit portrait. In this context, there is no assumption of hyperbolicity or connectedness of Julia sets, and so additional work must be done to show that a version of the uniformizing map can still be defined in this case. I prove that this is indeed the case, and establish a maximal subdomain of the basin of attraction at infinity on which this uniformizing map can be defined. I then show that even without an a priori assumption of hyperbolicity, the landing point of the rays specified by the orbit portrait is necessarily hyperbolic. I use this fact to prove that, similar to the previous case, the rays and landing point for the orbit portrait move holomorphically as the parameter is varied within the parameter wake for the orbit portrait. Finally, I prove several results characterizing the basic topology of the subsets of parameter space in question.^
Todd P Woodard,
"Using external rays to classify parameter space for non-autonomous polynomial dynamical systems"
Dissertations and Master's Theses (Campus Access).