| The dynamics of singularly perturbed complex rational maps is explored. These rational maps are of the form flambda(z) = zn + lambda/ &parl0;&parl0;z-a&parr0;da&parl0;z-b&parr0; db&parr0; where n, da and db are integers such that n ≥ 2, da, d b ≥ 1 and a, b and lambda∈ C such that ∣a∣, ∣b∣ ≠ 1 ∣lambda∣ is sufficiently small. The topological characteristics of the Julia and Fatou sets of these maps are studied. The dynamics of these maps on their Julia sets are also described and modeled using symbolic dynamics. Despite the large number of possibilities we show that in most cases the Julia set of flambda consists of a countable number of disjoint simple closed curves and uncountably many point components that accumulate on each of these curves. The main differences appear in the topological structure of the Fatou set of the map for different positions and orders of the poles a and b. We show that the Fatou set may consist of the disjoint union of one infinitely connected component and countably many disks, every component of the Fatou set is infinitely connected, or some components of the Fatou set are infinitely connected, some are disks and some are annuli. |
