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Frontiers in Complex DynamicsIn Celebration of John Milnor's 80th Birthday$
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Araceli Bonifant, Misha Lyubich, and Scott Sutherland

Print publication date: 2014

Print ISBN-13: 9780691159294

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159294.001.0001

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Limiting behavior of Julia sets of singularly perturbed rational maps

Limiting behavior of Julia sets of singularly perturbed rational maps

(p.121) Limiting behavior of Julia sets of singularly perturbed rational maps
Frontiers in Complex Dynamics

Robert L. Devaney

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Princeton University Press

This chapter surveys dynamical properties of the families fsubscript c,𝜆(z) = zⁿ + c + λ‎/zᵈ for n ≥ 2, d ≥ 1, with c corresponding to the center of a hyperbolic component of the Multibrot set. These rational maps produce a variety of interesting Julia sets, including Sierpinski carpets and Sierpinski gaskets, as well as laminations by Jordan curves. The chapter describes a curious “implosion” of the Julia sets as a polynomial psubscript c = zⁿ + c is perturbed to a rational map fsubscript c,𝜆. In this way the chapter shows yet another way of producing rational maps through “singular” perturbations of complex polynomials.

Keywords:   Julia sets, limiting behavior, rational maps, Sierpinski carpets, Sierpinski gaskets, Jordan curves, laminations, implosions, singularly perturbed rational maps

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