Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-14T09:22:00.312Z 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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Information

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

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