Abstract
Let f be a hyperbolic rational map with degree \(d\ge 2\) whose Julia set is connected. We give an elementary approach to prove that there exists a rational map g with degree \(\le 7d-2\) such that g contains a buried Julia component which is homeomorphic to the Julia set of f.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability Statement
The authors declare that data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
Notes
If \(p_i=1\) for some \(0\le i\le n-2\), then \(O_{i,1}=O_{i,0}\) and \(\gamma _{i,1}\) is regarded as a curve in \(B_{i,0}\).
Note that the constant \(\delta \) depends on the integer \(m\ge 3\). In the rest of this section, we always assume that \(\delta \) is chosen such that \(f_\lambda \) satisfies all the results obtained in Sect. 2.
We will see later that \(h_1(\eta , J(g))\) is actually a Julia component of \(f_{\lambda ,\eta }\).
References
McMullen, C.T.: Automorphisms of rational maps. Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 31-60, Math. Sci. Res. Inst. Publ., 10, Springer, New York (1988)
Devaney, R.L., Look, D.M., Uminsky, D.: The escape trichotomy for singularly perturbed rational maps. Indiana Univ. Math. J. 54(6), 1621–1634 (2005)
Xiao, Y., Qiu, W., Yin, Y.: On the dynamics of generalized McMullen maps. Ergodic Theory Dynam. Syst. 34(6), 2093–2112 (2014)
Pilgrim, K., Tan, L.: Rational maps with disconnected Julia set. Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque No. 261, 349–384 (2000)
Qiao, J.: The buried points on the Julia sets of rational and entire functions. Sci. China Ser. A. 38(12), 1409–1419 (1995)
Blanchard, P., Devaney, R.L., Garijo, A., Russell, E.D.: A generalized version of the McMullen domain. Internat J. Bifur Chaos Appl. Sci. Eng. 18(8), 2309–2318 (2008)
Godillon, S.: A family of rational maps with buried Julia components. Ergodic Theory Dynam. Syst. 35(6), 1846–1879 (2015)
Wang, Y., Yang, F.: Julia sets as buried Julia components. Trans. Am. Math. Soc. 373(10), 7287–7326 (2020)
Shishikura, M.: On the quasiconformal surgery of rational functions. Ann. Sci. École Norm. Sup. 20(4), 1–29 (1987)
Beardon, A.F.: Iteration of rational functions. Complex analytic dynamical systems. Graduate Texts in Mathematics. 132, Springer-Verlag, New York (1991)
Garijo, A., Marotta, S.M., Russell, E.D.: Singular perturbations in the quadratic family with multiple poles. J. Difference Equ. Appl. 19(1), 124–145 (2013)
Milnor, J: Dynamics in one complex variable: Third Edition. Annals of Mathematics Studies, Princeton Univ Press, Princeton (2006)
Mañé, R., Sad, P., Sullivan, D.: On the dynamics of rational maps. Ann. Sci. École Norm. Sup. 16(2), 193–217 (1983)
McMullen, C.T.: Complex dynamics and renormalization. Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ (1994)
Acknowledgements
The authors would like to thank Yang Fei for many helpful discussions and for a detailed reading of a draft. This work is supported by the NSFC (Grant Nos. 11871208, 11601481) and by Hunan Provincial Natural Science Foundation of China (Grant Nos. 2021JJ30308, 2018JJ2159.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wang, Y., Zhan, G. & Liao, L. Buried Julia Components and Julia Sets. Qual. Theory Dyn. Syst. 21, 22 (2022). https://doi.org/10.1007/s12346-021-00550-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-021-00550-1