Hostname: page-component-77c78cf97d-kmjgn Total loading time: 0.001 Render date: 2026-04-24T00:35:50.056Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Branner, B. and Hubbard, J. H.. The iteration of cubic polynomials, part I: the global topology of parameter space. Acta Math. 160 (1988), 143–206.CrossRefGoogle Scholar
Basu, S., Pollack, R. and Coste-Roy, M.-F.. Algorithms in Real Algebraic Geometry, 2nd edn. Springer, Berlin, 2006.Google Scholar
Bruin, H. and Schleicher, D.. Symbolic Dynamics of Quadratic Polynomials. Tech. Report 7. Institut Mittag-Leffler, available at: https://www.mittag-leffler.se/preprints/files/IML-0102-07.pdf.Google Scholar
Carleson, L. and Gamelin, T. W.. Complex Dynamics. Springer, Berlin, 1993.CrossRefGoogle Scholar
Crowe, W. D., Hasson, R., Rippon, P. J. and Strain-Clark, P. E. D.. On the structure of the Mandelbar set. Nonlinearity 2 (1989), 541–553.Google Scholar
Douady, A.. Systèmes dynamiques holomorphes. Seminaire Bourbaki, Astérisque 599 (1983), 39–63.Google Scholar
Douady, A. and Hubbard, J. H.. On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér. 18 (1985), 287–343.CrossRefGoogle Scholar
Douady, A. and Hubbard, J. H.. Étude Dynamique Des Polynômes Complexes I. Publications Mathématiques d’Orsay, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984–1985.Google Scholar
Eberlein, D., Mukherjee, S. and Schleicher, D.. Rational parameter rays of Multibrot sets, 2014, in preparation.Google Scholar
Goldberg, L. R. and Milnor, J.. Fixed points of polynomial maps II: Fixed point portraits. Ann. Sci. Éc. Norm. Supér. (4) 26 (1993), 51–98.Google Scholar
Hubbard, J. and Schleicher, D.. Multicorns are not path connected. Frontiers in Complex Dynamics: In Celebration of John Milnor’s 80th Birthday. Princeton University Press, Princeton, NJ, 2014, pp. 73–102.Google Scholar
Inou, H. and Mukherjee, S.. Non-landing parameter rays of the multicorns, submitted, 2014 http://arxiv.org/pdf/1406.3428v2.pdf.Google Scholar
Lau, E. and Schleicher, D.. Symmetries of fractals revisited. Math. Intelligencer 18 (1996), 45–51.CrossRefGoogle Scholar
Milnor, J.. Remarks on iterated cubic maps. Exp. Math. 1 (1992), 1–24.Google Scholar
Milnor, J.. Periodic orbits, external rays and the Mandelbrot set. Astèrisque 261 (2000), 277–333.Google Scholar
Milnor, J.. Dynamics in One Complex Variable, 3rd edn. Princeton University Press, NJ, 2006.Google Scholar
Mané, R., Sad, P. and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4) 16(2) (1983), 193–217.Google Scholar
Mukherjee, S.. Orbit portraits of unicritical antiholomorphic polynomials. Conform. Geom. Dyn. 19 (2015), 35–50.CrossRefGoogle Scholar
Mukherjee, S.. Real-analyticity of Hausdorff dimension of Julia sets along the parabolic arcs of the multicorns, submitted, 2014 http://arxiv.org/abs/1410.1180v1.Google Scholar
Nakane, S.. Connectedness of the Tricorn. Ergod. Th. & Dynam. Sys. 13 (1993), 349–356.Google Scholar
Nakane, S.. On quasiconformal equivalence on the boundary of the Tricorn. Structure and Bifurcation of Dynamical Systems (Advanced Series in Dynamical System, 11) . World Scientific, Hackensack, NJ, 1992, pp. 154–167.Google Scholar
Nakane, S. and Schleicher, D.. On multicorns and unicorns I: antiholomorphic dynamics, hyperbolic components and real cubic polynomials. Internat. J. Bifur. Chaos 13 (2003), 2825–2844.CrossRefGoogle Scholar
Poirier, A.. On postcritically finite polynomials, Part one: Critical portraits. Preprint, 1993,arXiv:9305207.Google Scholar
Schleicher, D.. Rational parameter rays of the Mandelbrot set. Astérisque 261 (2000), 405–443.Google Scholar
Zinsmeister, M.. Thermodynamic Formalism and Holomorphic Dynamical Systems (SMF/AMS Texts and Monographs, 2) . AMS and Société Mathématique de France, Marseille, 2000.Google Scholar