Dynamics of nonholomorphic singular continuations: A case in radial symmetry

This paper is primarily a study of special families of rational maps of the real plane of the form: z → zⁿ + β/zⁿ, where the dynamic variable z ∈ C, and C is identified with R². The parameter β is complex; n is a positive integer. For β small, this family can be considered a nonholomorphic singular perturbation of the holomorphic family z → zⁿ, although we consider large values of β as well. Compared to the more general family z → zⁿ + c + β/2^d, the special case where n = d and c = 0 is easier to analyze because the radial component in polar coordinates decouples from the angular component. This reduces a significant part of the analysis to the study of a family of one-real-dimensional unimodal maps. For each fixed n, the β parameter plane separates into three major regions, corresponding to maps which have one of the following behaviors: (i) all orbits go off to infinity, (ii) only an annulus of points stays bounded, and (iii) only a Cantor set of circles stays bounded. In cases (ii) and (iii), there is a transitive invariant set; this set is an attractor in case (ii). Some comparisons are made between z ⁿ + β/zⁿ and the holomorphic singularly perturbed maps: z → zⁿ + λ/zⁿ, studied by Devaney and coauthors over the last decade. Additional results and observations are made about the more general family where c ≠ 0 and n ≠ d.