Skip to main content
×
Home
    • Aa
    • Aa

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

  • SABYASACHI MUKHERJEE (a1), SHIZUO NAKANE (a2) and DIERK SCHLEICHER (a1)
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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

B. Branner and J. H. Hubbard . The iteration of cubic polynomials, part I: the global topology of parameter space. Acta Math. 160 (1988), 143206.

L. Carleson and T. W. Gamelin . Complex Dynamics. Springer, Berlin, 1993.

W. D. Crowe , R. Hasson , P. J. Rippon and P. E. D. Strain-Clark . On the structure of the Mandelbar set. Nonlinearity 2 (1989), 541553.

A. Douady and J. H. Hubbard . On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér. 18 (1985), 287343.

L. R. Goldberg and J. Milnor . Fixed points of polynomial maps II: Fixed point portraits. Ann. Sci. Éc. Norm. Supér. (4) 26 (1993), 5198.

E. Lau and D. Schleicher . Symmetries of fractals revisited. Math. Intelligencer 18 (1996), 4551.

R. Mané , P. Sad and D. Sullivan . On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4) 16(2) (1983), 193217.

S. Mukherjee . Orbit portraits of unicritical antiholomorphic polynomials. Conform. Geom. Dyn. 19 (2015), 3550.

S. Nakane . Connectedness of the Tricorn. Ergod. Th. & Dynam. Sys. 13 (1993), 349356.

S. Nakane and D. Schleicher . On multicorns and unicorns I: antiholomorphic dynamics, hyperbolic components and real cubic polynomials. Internat. J. Bifur. Chaos 13 (2003), 28252844.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 25 *
Loading metrics...

Abstract views

Total abstract views: 345 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th September 2017. This data will be updated every 24 hours.