Skip to main content
SpringerLink
Log in

Quadratic Rational Maps with a 2-Cycle of Siegel Disks

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

For the family of quadratic rational functions having a 2-cycle of bounded type Siegel disks, we prove that each of the boundaries of these Siegel disks contains at most one critical point. In the parameter plane, we prove that the locus for which the boundaries of the 2-cycle of Siegel disks contain two critical points is a Jordan curve.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Aspenberg, M., Yampolsky, M.: Mating non-renormalizable quadratic polynomials. Commun. Math. Phys. 287(1), 1–40 (2009)

    Article  MathSciNet  Google Scholar 

  2. Blokh, A., Chéritat, A., Oversteegen, L., Timorin, V.: Location of Siegel capture polynomials in parameter spaces. Nonlinearity 34(4), 2430–2453 (2021)

    Article  MathSciNet  Google Scholar 

  3. Buff, X., Écalle, J., Epstein, A.: Limits of degenerate parabolic quadratic rational maps. Geom. Funct. Anal. 23(1), 42–95 (2013)

    Article  MathSciNet  Google Scholar 

  4. Berenguel, R., Fagella, N.: An entire transcendental family with a persistent Siegel disc. J. Differ. Equ. Appl. 16(5–6), 523–553 (2010)

    Article  MathSciNet  Google Scholar 

  5. Buff, X., Henriksen, C.: Julia sets in parameter spaces. Commun. Math. Phys. 220(2), 333–375 (2001)

    Article  MathSciNet  Google Scholar 

  6. Chéritat, A.: On the size of Siegel disks with fixed multiplier for cubic polynomials, arXiv:2003.13337 (2020)

  7. Cheraghi, D.: Topology of irrationally indifferent attractors, arXiv:1706.02678v3 (2022)

  8. DeMarco, L.: Iteration at the boundary of the space of rational maps. Duke Math. J. 130(1), 169–197 (2005)

    Article  MathSciNet  Google Scholar 

  9. DeMarco, L.: The moduli space of quadratic rational maps. J. Am. Math. Soc. 20(2), 321–355 (2007)

    Article  MathSciNet  Google Scholar 

  10. Devaney, R.L., Fagella, N., Garijo, A., Jarque, X.: Sierpiński curve Julia sets for quadratic rational maps. Ann. Acad. Sci. Fenn. Math. 39(1), 3–22 (2014)

    Article  MathSciNet  Google Scholar 

  11. Epstein, A.L.: Bounded hyperbolic components of quadratic rational maps. Ergodic Theory Dynam. Systems 20(3), 727–748 (2000)

    Article  MathSciNet  Google Scholar 

  12. Graczyk, J., Świa̧tek, G.: Siegel disks with critical points in their boundaries. Duke Math. J. 119(1), 189–196 (2003)

    Article  MathSciNet  Google Scholar 

  13. Keen, L., Zhang, G.: Bounded-type Siegel disks of a one-dimensional family of entire functions. Ergodic Theory Dynam. Syst. 29(1), 137–164 (2009)

    Article  MathSciNet  Google Scholar 

  14. Lehto, O.: Univalent Functions and Teichmüller Spaces. Graduate Texts in Mathematics, vol. 109. Springer, New York (1987)

    Book  Google Scholar 

  15. Lyubich, M.: Some typical properties of the dynamics of rational maps. Russian Math. Surv. 38(5), 154–155 (1983)

    Article  Google Scholar 

  16. McMullen, C.T.: Complex Dynamics and Renormalization. Annals of Mathematics Studies, vol. 135. Princeton University Press, Princeton (1994)

    MATH  Google Scholar 

  17. McMullen, C.T.: Self-similarity of Siegel disks and Hausdorff dimension of Julia sets. Acta Math. 180(2), 247–292 (1998)

    Article  MathSciNet  Google Scholar 

  18. Milnor, J.: Geometry and dynamics of quadratic rational maps, with an appendix by the author and Lei Tan. Exp. Math. 2(1), 37–83 (1993)

    Article  Google Scholar 

  19. Milnor, J.: Dynamics in One Complex Variable. Annals of Mathematics Studies, vol. 3. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  20. Petersen, C.L.: Local connectivity of some Julia sets containing a circle with an irrational rotation. Acta Math. 177(2), 163–224 (1996)

    Article  MathSciNet  Google Scholar 

  21. Petersen, C.L., Zakeri, S.: On the Julia set of a typical quadratic polynomial with a Siegel disk. Ann. Math. 159(1), 1–52 (2004)

    Article  MathSciNet  Google Scholar 

  22. Rees, M.: Components of degree two hyperbolic rational maps. Invent. Math. 100(2), 357–382 (1990)

    Article  MathSciNet  Google Scholar 

  23. Shishikura, M., Yang, F.: The high type quadratic Siegel disks are Jordan domains, arXiv:1608.04106v4, (2021)

  24. Yang, F.: On the dynamics of a family of entire functions. Acta Math. Sin. (Engl. Ser.) 29(11), 2047–2072 (2013)

    Article  MathSciNet  Google Scholar 

  25. Zakeri, S.: Dynamics of cubic Siegel polynomials. Commun. Math. Phys. 206(1), 185–233 (1999)

    Article  MathSciNet  Google Scholar 

  26. Zakeri, S.: On Siegel disks of a class of entire maps. Duke Math. J. 152(3), 481–532 (2010)

    Article  MathSciNet  Google Scholar 

  27. Zakeri, S.: Rotation Sets and Complex Dynamics. Lecture Notes in Mathematics, vol. 2214. Springer, Cham (2018)

    Book  Google Scholar 

  28. Zhang, G.: On the non-escaping set of \(e^{2\pi i\theta }\sin (z)\). Isr. J. Math. 165, 233–252 (2008)

    Article  Google Scholar 

  29. Zhang, G.: All bounded type Siegel disks of rational maps are quasi-disks. Invent. Math. 185(2), 421–466 (2011)

    Article  MathSciNet  Google Scholar 

  30. Zhang, G.: Topological characterisation of rational maps with Siegel disks. Math. Proc. Camb. Philos. Soc. 172(1), 1–41 (2022)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Arnaud Chéritat for providing an algorithm to draw Fig. 1, and the referee for very insightful and detailed comments and corrections. This work was supported by NSFC (grant Nos. 12071210, 12171276) and NSF of Jiangsu Province (grant No. BK20191246).

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518061, People’s Republic of China

    Yuming Fu

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Fei Yang & Gaofei Zhang

Authors
  1. Yuming Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gaofei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fei Yang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fu, Y., Yang, F. & Zhang, G. Quadratic Rational Maps with a 2-Cycle of Siegel Disks. J Geom Anal 32, 244 (2022). https://doi.org/10.1007/s12220-022-00989-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-022-00989-x

Keywords

Mathematics Subject Classification

Search

Navigation