Invariant Jordan curves in the Julia sets of rational maps

Xiuming Zhang; Xiuming Zhang

doi:10.3934/math.2025747

AIMS Mathematics

2025, Volume 10, Issue 7: 16664-16675. doi: 10.3934/math.2025747

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Research article

Invariant Jordan curves in the Julia sets of rational maps

Xiuming Zhang ^,

School of Mathematics, Qilu Normal University, Jinan 250200, China

Received: 24 March 2025 Revised: 28 June 2025 Accepted: 08 July 2025 Published: 24 July 2025
MSC : 37F10, 37F20

We give a way to construct invariant Jordan curves in the Julia sets of rational maps that are not boundaries of Fatou components.
- Julia sets,
- invariant Jordan curves,
- mating,
- rational maps,
- combinatorial equivalence
Citation: Xiuming Zhang. Invariant Jordan curves in the Julia sets of rational maps[J]. AIMS Mathematics, 2025, 10(7): 16664-16675. doi: 10.3934/math.2025747

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Abstract

We give a way to construct invariant Jordan curves in the Julia sets of rational maps that are not boundaries of Fatou components.

References

[1]	P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France, 48 (1920), 208–314. https://doi.org/10.24033/bsmf.1008 doi: 10.24033/bsmf.1008
[2]	A. Douady, Disques de Siegel et anneaux de Herman, Astérisque, 152-153 (1987), 151–172.
[3]	M. R. Herman, Conjugaison quasi-symmétrique des homéomorphismes analytiques du cercle à des rotations, preliminary manuscript, 1987.
[4]	C. L. Petersen, S. Zakeri, On the Julia set of a typical quadratic polynomial with a Siegel disk, Ann. Math., 159 (2004), 1–52. https://doi.org/10.4007/annals.2004.159.1 doi: 10.4007/annals.2004.159.1
[5]	A. Avila, X. Buff, A. Chéritat, Siegel disks with smooth boundaries, Acta Math., 193 (2004), 1–30. https://doi.org/10.1007/bf02392549 doi: 10.1007/bf02392549
[6]	G. Zhang, All bounded type Siegel disks of rational maps are quasi-disks, Invent. Math., 185 (2011), 421–466. https://doi.org/10.1007/s00222-011-0312-0 doi: 10.1007/s00222-011-0312-0
[7]	M. Shishikura, F. Yang, The high type quadratic Siegel disks are Jordan domains, J. Eur. Math. Soc., 2024. https://doi.org/10.4171/jems/1481
[8]	D. Cheraghi, Topology of irrationally indifferent attractors, arXiv: 1706.02678, 2025. https://doi.org/10.48550/arXiv.1706.02678
[9]	M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1–29. https://doi.org/10.24033/asens.1522 doi: 10.24033/asens.1522
[10]	P. Roesch, Y. Yin, Bounded critical Fatou components are Jordan domains for polynomials, Sci. China Math., 65 (2022), 331–358. https://doi.org/10.1007/s11425-020-1827-4 doi: 10.1007/s11425-020-1827-4
[11]	A. Eremenko, Invariant curves and semiconjugacies of rational functions, Fund. Math., 219 (2012), 263–270. https://doi.org/10.4064/fm219-3-5 doi: 10.4064/fm219-3-5
[12]	F. Yang, Rational maps with smooth degenerate Herman rings, Adv. Math., 452 (2024), 1–29. https://doi.org/10.1016/j.aim.2024.109827 doi: 10.1016/j.aim.2024.109827
[13]	W. R. Lim, A priori bounds and degeneration of Herman rings with bounded type rotation number, arXiv: 2302.07794, 2023. https://doi.org/10.48550/arXiv.2302.07794
[14]	G. Zhang, Jordan mating is always possible for polynomials, Math. Z., 306 (2024), 1–10. https://doi.org/10.1007/s00209-024-03465-0 doi: 10.1007/s00209-024-03465-0
[15]	L. Tan, Matings of quadratic polynomials, Ergod. Theor. Dyn. Syst., 12 (1992), 589–620. https://doi.org/10.1017/s0143385700006957 doi: 10.1017/s0143385700006957
[16]	A. Douady, J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263–297. https://doi.org/10.1007/bf02392534 doi: 10.1007/bf02392534
[17]	B. Bielefeld, Y. Fisher, J. Hubbard, The classification of critically preperiodic polynomials as dynamical systems, J. Amer. Math. Soc., 5 (1992), 721–762. https://doi.org/10.1090/s0894-0347-1992-1149891-3 doi: 10.1090/s0894-0347-1992-1149891-3
[18]	K. M. Pilgrim, Canonical Thurston obstructions, Adv. Math., 158 (2001), 154–168. https://doi.org/10.1006/aima.2000.1971 doi: 10.1006/aima.2000.1971
[19]	J. H. Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 2, Surface homeomorphisms and rational functions, New York: Matrix Editions, 2016. https://doi.org/hal-01297628
[20]	D. P. Thurston, A positive characterization of rational maps, Ann. Math., 192 (2020), 1–46. https://doi.org/10.4007/annals.2020.192.1.1 doi: 10.4007/annals.2020.192.1.1
[21]	J. Milnor, Dynamics in one complex variable, 3 Eds., Princeton University Press, 2006. http://dx.doi.org/10.1177/0300060513509036
[22]	L. Tan, Y. Yin, Local connectivity of the Julia set for geometrically finite rational maps, Sci. China Ser. A, 39 (1996), 39–47.

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