Connectivity of Julia Sets for Singularly Perturbed Rational Maps
Abstract
In this paper we consider the family of rational maps of the form Fλ(z) = zn + λ/zn where n ≥ 2. It is known that there are two cases where the Julia sets of these maps are not connected. If the critical values of Fλ lie in the basin of ∞, then the Julia set is a Cantor set. And if the critical values lie in the preimage of the basin surrounding the pole at 0, then the Julia set is a Cantor set of concentric simple closed curves around the origin. We prove in this paper that, in all other cases, the Julia set of Fλ is a connected set.
- Publication:
-
Chaos
- Pub Date:
- January 2013
- DOI:
- 10.1142/9789814434805_0018
- Bibcode:
- 2013ccmb.conf..239D