Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-17T01:56:50.834Z Has data issue: false hasContentIssue false
  • English
  • Français

Density of hyperbolicity for rational maps with Cantor Julia sets

Published online by Cambridge University Press:  08 September 2011

WENJUAN PENG ,
YONGCHENG YIN  and
YU ZHAI
WENJUAN PENG
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, PR China (email: wenjpeng@amss.ac.cn)
YONGCHENG YIN
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, PR China (email: yin@zju.edu.cn)
YU ZHAI
Affiliation:
Department of Mathematics, School of Science, China University of Mining and Technology (Beijing), Beijing 100083, PR China (email: zhaiyu@amss.ac.cn)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, taking advantage of quasi-conformal surgery, we prove that each non-hyperbolic rational map with a Cantor Julia set can be approximated by hyperbolic rational maps with Cantor Julia sets of the same degree.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5 , October 2012 , pp. 1711 - 1726
Copyright
Copyright © Cambridge University Press 2011

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

[Ah]Ahlfors, L.. Lectures on Quasiconformal Mappings, 2nd edn.(University Lecture Series, 38). American Mathematical Society, Providence, RI, 2006, with additional chapters by: C. J. Earle and I. Kra, M. Shishikura and J. H. Hubbard.CrossRefGoogle Scholar
[BH]Branner, B. and Hubbard, J. H.. The iteration of cubic polynomials, Part II: patterns and parapatterns. Acta Math. 169 (1992), 229–325.CrossRefGoogle Scholar
[Cui]Cui, G.. Dynamics of rational maps, topology, deformation and bifurcation. Preprint, 2002.Google Scholar
[CP]Cui, G. and Peng, W.. On the structure of Fatou domains. Sci. China Ser. A-Math. 51 (2008), 1167–1186.CrossRefGoogle Scholar
[DH]Douady, A. and Hubbard, J. H.. On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér. 18 (1985), 287–344.CrossRefGoogle Scholar
[GS]Graczyk, J. and Świa̧tek, G.. Generic hyperbolicity in the logistic family. Ann. of Math. (2) 146 (1997), 1–52.CrossRefGoogle Scholar
[KL]Kahn, J. and Lyubich, M.. The quasi-additivity law in conformal geometry. Ann. of Math. (2) 169 (2009), 561–593.CrossRefGoogle Scholar
[KS]Kozlovski, O. and van Strien, S.. Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials. Proc. Lond. Math. Soc. 99 (2009), 275–296.CrossRefGoogle Scholar
[KSS]Kozlovski, O., Shen, W. and van Strien, S.. Rigidity for real polynomials. Ann. of Math. (2) 165 (2007), 749–841.CrossRefGoogle Scholar
[L]Lyubich, M.. Dynamics of quadratic polynomials, I–II. Acta Math. 178 (1997), 185–247, 247–297.CrossRefGoogle Scholar
[Mil]Milnor, J.. Dynamics in One Complex Variable: Introductory Lectures, 3rd edn.(Annals of Mathematics Studies, 160). Princeton University Press, Princeton, 2006.Google Scholar
[QY]Qiu, W. and Yin, Y.. Proof of the Branner–Hubbard conjecture on Cantor Julia sets. Sci. China Ser. A-Math. 52 (2009), 45–65.CrossRefGoogle Scholar
[Sh]Shishikura, M.. On the quasiconformal surgery of rational functions. Ann. Sci. Éc. Norm. Supér. 20 (1987), 1–29.CrossRefGoogle Scholar
[Su]Sullivan, D.. Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou–Julia problem on wandering domains. Ann. of Math. (2) 122 (1985), 401–418.CrossRefGoogle Scholar
[YZ]Yin, Y. and Zhai, Y.. No invariant line fields on Cantor Julia sets. Forum Math. 22 (2010), 75–94.CrossRefGoogle Scholar
[Zh1]Zhai, Y.. A generalized version of Branner–Hubbard conjecture for rational functions. Acta Math. Sinica 26 (2010), 2199–2208.CrossRefGoogle Scholar
[Zh2]Zhai, Y.. Rigidity for rational maps with Cantor Julia sets. Sci. China Ser. A-Math. 51 (2008), 79–92.CrossRefGoogle Scholar