A Generalized Version of the McMullen Domain

We study the family of complex maps given by Fλ(z) = zn + λ/zn + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components. ∗Dep. d’Eng. Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Päısos Catalans, 26, 43007 Tarragona, Spain †The third author would like to thank to Department of Mathematics at Boston University for their hospitality while this work was in progress. The third author was supported by MTM2005-02139/Consolider (including a FEDER contribution) and CIRIT 2005 SGR01028. ‡Please address all correspondence to Elizabeth D. Russell, Department of Mathematics, Boston University, 111 Cummington Street, Boston MA 02215, or e-mail [email protected].