A Connection between Analytic and Nonanalytic Singular Perturbations of the Quadratic Map

Abstract

To explore the connection between the analytic and the nonanalytic complex dy- namics, this paper studied a special case of the singularly perturbed quadratic map: β β ƒβ‚t (z) = z2 + t — + (1 – t) — z2 – z2 which can transit from nonanalytic to analytic by varying t from 0 to 1. Other variables involved in this map are the dynamic variable z ϵ C and the main parameter β ϵ R. Our investigation shows that this transition map has much richer behaviors than the end point cases. The parameter space can be no longer subdivided into only four or three regions as shown in the studies by Devaney and Bozyk respectively. Correspondingly, in the dynamic plane, there also appear several new intermediate cases among which we identified two transitions: a connected case where the filled Julia set is connected and a disconnected case where the filled Julia set consists of infinitely many components. This paper also pointed out that ƒβ‚t (z) is semiconjugate to the four symbols shift map for the disconnected case. This semiconjugacy provides a way to largely understand the dynamical behaviors for the non escape points. Further study shows that the critical set plays an important role in the construction of the filled Julia set. Therefore, the transition of the critical set and its image set are also discussed in this paper. At the end, we presented two sets of conjectures for the bounded critical orbits and the escape critical orbits for future study.