From Newton's method to exotic basins Part II: Bifurcation of the Mandelbrot-like sets

Volume 168 / 2001

Krzysztof Bara/nski Fundamenta Mathematicae 168 (2001), 1-55 MSC: Primary 37F45. DOI: 10.4064/fm168-1-1

Abstract

This is a continuation of the work [Ba] dealing with the family of all cubic rational maps with two supersinks. We prove the existence of the following parabolic bifurcation of Mandelbrot-like sets in the parameter space of this family. Starting from a Mandelbrot-like set in cubic Newton maps and changing parameters in a continuous way, we construct a path of Mandelbrot-like sets ending in the family of parabolic maps with a fixed point of multiplier $1$. Then it bifurcates into two paths of Mandelbrot-like sets, contained respectively in the set of maps with exotic or non-exotic basins. The non-exotic path ends at a Mandelbrot-like set in cubic polynomials.

Authors

  • Krzysztof Bara/nskiInstitute of Mathematics
    Warsaw University
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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