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On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials

Published online by Cambridge University Press:  27 November 2015

SABYASACHI MUKHERJEE Open the ORCID record for SABYASACHI MUKHERJEE [Opens in a new window] ,
SHIZUO NAKANE  and
DIERK SCHLEICHER
SABYASACHI MUKHERJEE
Affiliation:
Jacobs University Bremen, Campus Ring 1, Bremen, 28759, Germany email s.mukherjee@jacobs-university.de, sabyasachi.mukherjee@stonybrook.edu, d.schleicher@jacobs-university.de
SHIZUO NAKANE
Affiliation:
Tokyo Polytechnic University, 1583, Iiyama, Atsugi, Kanagawa 243-0297, Japan email nakane@gen.t-kougei.ac.jp
DIERK SCHLEICHER
Affiliation:
Jacobs University Bremen, Campus Ring 1, Bremen, 28759, Germany email s.mukherjee@jacobs-university.de, sabyasachi.mukherjee@stonybrook.edu, d.schleicher@jacobs-university.de
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Abstract

The multicorns are the connectedness loci of unicritical antiholomorphic polynomials $\bar{z}^{d}+c$. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio 2. We also count the number of hyperbolic components of any period of the multicorns. Since antiholomorphic polynomials depend only real-analytically on the parameters, most of the techniques used in this paper are quite different from the ones used to prove the corresponding results in a holomorphic setting.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 3 , May 2017 , pp. 859 - 899
Copyright
© Cambridge University Press, 2015 

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