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Singular perturbations of complex polynomials

Author: Robert L. Devaney
Journal: Bull. Amer. Math. Soc. 50 (2013), 391-429
MSC (2010): Primary 37F10; Secondary 37F45
Published electronically: April 2, 2013
MathSciNet review: 3049870
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Abstract: In this paper we describe the dynamics of singularly perturbed complex polynomials. That is, we start with a complex polynomial whose dynamics are well understood. Then we perturb this map by adding a pole, i.e., by adding in a term of the form $ \lambda /(z-a)^d$ where the parameter $ \lambda $ is complex. This changes the polynomial into a rational map of higher degree and, as we shall see, the dynamical behavior explodes.

One aim of this paper is to give a survey of the many different topological structures that arise in the dynamical and parameter planes for these singularly perturbed maps. We shall show how Sierpiński curves arise in a myriad of different ways as the Julia sets for these singularly perturbed maps, and while these sets are always the same topologically, the dynamical behavior on them is often quite different. We shall also describe a number of interesting topological objects that arise in the parameter plane (the $ \lambda $-plane) for these maps. These include Mandelpinski necklaces, Cantor webs, and Cantor sets of circles of Sierpiński curve Julia sets.

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Additional Information

Robert L. Devaney
Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts 02215

Received by editor(s): June 12, 2012
Published electronically: April 2, 2013
Additional Notes: The authors work was partially supported by grant #208780 from the Simons Foundation
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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