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

Krzysztof Bara/nski¹

¹ Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland

Fundamenta Mathematicae, Tome 168 (2001) no. 1, pp. 1-55

Résumé

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.

Détail

DOI : 10.4064/fm168-1-1

Keywords: continuation work dealing family cubic rational maps supersinks prove existence following parabolic bifurcation mandelbrot like sets parameter space family starting mandelbrot like set cubic newton maps changing parameters continuous construct path mandelbrot like sets ending family parabolic maps fixed point multiplier bifurcates paths mandelbrot like sets contained respectively set maps exotic non exotic basins non exotic path ends mandelbrot like set cubic polynomials