Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T15:47:47.056Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcations of dynamic rays in complex polynomials of degree two

Published online by Cambridge University Press:  19 September 2008

Pau Atela
Pau Atela
Affiliation:
Program in Applied Mathematics, University of Colorado, Boulder, CO 80309-526, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In the study of bifurcations of the family of degree-two complex polynomials, attention has been given mainly to parameter values within the Mandelbrot set M (e.g., connectedness of the Julia set and period doubling). The reason for this is that outside M, the Julia set is at all times a hyperbolic Cantor set. In this paper weconsider precisely this, values of the parameter in the complement of M. We find bifurcations occurring not on the Julia set itself but on the dynamic rays landing on itfrom infinity. As the parameter crosses the external rays of M, in the dynamic plane the points of the Julia set gain and lose dynamic rays. We describe these bifurcations with the aid of a family of circle maps and we study in detail the case of the fixed points.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 3 , September 1992 , pp. 401 - 423
Copyright
Copyright © Cambridge University Press 1992

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.)

References

REFERENCES

[A]Atela, P.. The Mandelbrot set and σ-automorphisms of quotients of the shift. Trans. Amer. Math. Soc. to appear.Google Scholar
[B]Blanchard, P.. Complex analytic dynamics on the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85–141.CrossRefGoogle Scholar
[Bo]Boyland, P.. Bifurcations of circle Maps: Arnol'd tongues, bistability and rotation intervals. Commun. Math. Phys. 106 (1986), 353–381.CrossRefGoogle Scholar
[Br]Branner, B.. The Mandelbrot set. In: Chaos and Fractals, Proc. Symp. in Applied Mathematics, 39, Devaney, R. and Keen, L., eds, Amer. Math. Soc.; 1989.Google Scholar
[D]Devaney, R.. An Introduction to Chaotic Dynnamical Systems. Addison Wesley, 2nd ed., 1989.Google Scholar
[DH]Douady, A. & Hubbard, J. H.. On the dynamics of polynomial- like mappings. Ann. Sci. École Norm. Sup. 18 (1985), 287–343.CrossRefGoogle Scholar
[DH1]Douady, A. & Hubbard, J. H.. Itération des polynômes quadratiques complexes. CR. Acad. Sci. Paris Série I 294, (1982), 123–126.Google Scholar
[DH2]Douady, A. & Hubbard, J. H.. Etude Dynamique des Polynômes Complexes. Publ. Math. d'Orsay, Part I, No. 84–02, (1984), and Part II, No. 85–04, (1985).Google Scholar
[Dou]Douady, A.. Algorithms for computing angles in the Mandelbrot set. Chaotic Dynamics and Fractals. Barnsley, M. and Demko, S. G., eds, Academic Press: New York, 1986, 155–168.CrossRefGoogle Scholar
[G-M]Goldberg, L. & Milnor, J.. Fixed point portraints of polynomial maps. Preprint.Google Scholar
[H]Hall, G.. A C∞ Denjoy counterexample. Ergod. Th. & Dynam. Sys. 1 (1981), 261–272.CrossRefGoogle Scholar
[He]Herman, M.. Sur la conjugaison diffélrentiable des difieomorphismes du cercle à des rotations. Publ. Math., IHES 49 (1979), 5–234.CrossRefGoogle Scholar
[K]Linda, Keen. Julia sets. In: Chaos and Fractals, Proc Symp. in Applied Mathematics, 39, Devaney, R and Keen, L., eds, Amer. Math. Soc.: 1989.Google Scholar