Singular perturbations of complex analytic dynamical systems

Abstract

Our goal in this paper is to describe the dynamical behavior of singular perturbations of complex dynamical systems. Singular perturbations occur when a pole is introduced into the dynamics of a polynomial. In this paper, we consider the simplest possible case: start with the map Z → Zn where n > 1 and then add a pole at the origin. For simplicity, we consider the case of maps of the form Zn +λ/Zn. We then describe some of the many ways Sierpinski curve Julia sets arise in this family. We also give a classification of the dynamics on these sets and describe the intricate structure that occurs around the McMullen domain in the parameter plane for these maps. Finally, we discuss some of the major differences between the cases n = 2 and n > 2. © 2010 Springer-Verlag Berlin Heidelberg.