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Connectivity of the Julia sets of singularly perturbed rational maps

  • Jianxun FuEmail author
  • Yanhua Zhang
Article
  • 26 Downloads

Abstract

We consider a family of rational functions which is given by
$$\begin{aligned} f_{\lambda }(z)=\frac{z^n(z^{2n}-\lambda ^{n+1})}{z^{2n}-\lambda ^{3n-1}}, \end{aligned}$$
where \(n\ge 2\) and \(\lambda \in {\mathbb {C}}^*-\{\lambda :\lambda ^{2n-2}=1\}\). When \(\lambda \ne 0\) is small, \(f_{\lambda }\) can be seen as a perturbation of the unicritical polynomial \(z\mapsto z^n\). It was known that in this case the Julia set \(J(f_\lambda )\) of \(f_\lambda \) is a Cantor set of circles on which the dynamics of \(f_\lambda \) is not topologically conjugate to that of any McMullen maps. In this paper, we prove that this is the unique case such that \(J(f_\lambda )\) is disconnected.

Keywords

Julia sets connectivity Herman ring Cantor circles 

2010 Mathematics Subject Classification

Primary: 37F45 Secondary: 37F10 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11401298). The authors would like to thank Fei Yang and Gaofei Zhang for helpful suggestions and encouragements.

References

  1. 1.
    Beardon A, Iteration of Rational Functions, Graduate Texts in Mathematics 132 (1991) (New York: Springer-Verlag)Google Scholar
  2. 2.
    Carleson L and Gamelin T, Complex Dynamics (1993) (New York: Springer-Verlag)CrossRefGoogle Scholar
  3. 3.
    Devaney R and Russell E, Connectivity of Julia sets for singularly perturbed rational maps, Chaos, CNN, Memristors and Beyond (2013) (World Scientific) pp. 239–245CrossRefGoogle Scholar
  4. 4.
    Fu J and Yang F, On the dynamics of a family of singularly perturbed rational maps, J. Math. Anal. Appl. 424 (2015) 104–121MathSciNetCrossRefGoogle Scholar
  5. 5.
    McMullen C T, Automorphisms of rational maps, in: Holomorphic Functions and Moduli I, Math. Sci. Res. Inst. Publ. (1988) (Springer) vol. 10Google Scholar
  6. 6.
    Milnor J, Dynamics in One Complex Variable, third edition (2006) (Princeton: Princeton Univ. Press)zbMATHGoogle Scholar
  7. 7.
    Qiao J and Li Y, On connectivity of Julia sets of Yang–Lee zeros, Comm. Math. Phys. 222(2) (2001) 319–326MathSciNetCrossRefGoogle Scholar
  8. 8.
    Qiu W, Yang F and Yin Y, Rational maps whose Julia sets are Cantor circles, Ergod. Th. Dynam. Sys. 35 (2015) 499–529MathSciNetCrossRefGoogle Scholar
  9. 9.
    Qiu W, Yang F and Yin Y, Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps, Discret. Contin. Dyn. Syst. 36 (2016) 3375–3416MathSciNetCrossRefGoogle Scholar
  10. 10.
    Shishikura M, The connectivity of the Julia set and fixed points, in: Complex dynamics, edited by A K Peters (2009) (Wellesley, MA) pp. 257–276Google Scholar
  11. 11.
    Whyburn G T, Topological characterization of the Sierpiński curve, Fund. Math. 45 (1958) 320–324MathSciNetCrossRefGoogle Scholar
  12. 12.
    Xiao Y and Qiu W, The rational maps \(F_\lambda (z)=z^m+\lambda /z^d\) have no Herman rings, Proc. lndian Acad. Sci. (Math. Sci.) 120 (2010) 403–407CrossRefGoogle Scholar
  13. 13.
    Yang F, Rational maps without Herman rings, Proc. Amer. Math. Soc. (Math. Sci.) 145(4) (2017) 1649–1659MathSciNetzbMATHGoogle Scholar
  14. 14.
    Yin Y, On the Julia sets of quadratic rational maps, Complex Variables Theory Appl. 18(3–4) (1992) 141–147MathSciNetzbMATHGoogle Scholar
  15. 15.
    Yang F and Zeng J, On the dynamics of a family of generated renormalization transformations, J. Math. Anal. Appl. 413(1) (2014) 361–377MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.Department of MathematicsQufu Normal UniversityQufuPeople’s Republic of China

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