Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-13T00:02:03.755Z Has data issue: false hasContentIssue false
  • English
  • Français

Orbifold expansion and entire functions with bounded Fatou components

Part of:Entire and meromorphic functions, and related topicsComplex dynamical systemsDifferential topology

Published online by Cambridge University Press:  04 March 2021

LETICIA PARDO-SIMÓNOpen the ORCID record for LETICIA PARDO-SIMÓN[Opens in a new window]
LETICIA PARDO-SIMÓN*
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland
*
e-mail: l.pardo-simon@impan.pl
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.

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

Keywords

hyperbolic orbifoldentire functionFatou setlocal connectivityEremenko– Lyubich class

MSC classification

Primary:37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary:30D05: Functional equations in the complex domain, iteration and composition of analytic functions57R18: Topology and geometry of orbifolds

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 5 , May 2022, pp. 1807 - 1846
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Abramowitz, M. and Stegun, I.A.. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. US Department of Commerce, Washington, DC, 1972.Google Scholar
Baker, I. N.. The domains of normality of an entire function. Ann. Acad. Sci. Fenn. Ser. A I 1 (1975), 277–283.Google Scholar
Baker, I. N.. Wandering domains in the iteration of entire functions. Proc. Lond. Math. Soc. (3) 49 (1984), 563–576.CrossRefGoogle Scholar
Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29(2) (1993), 151–188.CrossRefGoogle Scholar
Bergweiler, W., Fagella, N. and Rempe-Gillen, L.. Hyperbolic entire functions with bounded Fatou components. Comment. Math. Helv. 90(4) (2015), 799–829.CrossRefGoogle Scholar
Bergweiler, W., Haruta, M., Kriete, H., Meier, H.-G. and Terglane, N.. On the limit functions of iterates in wandering domains. Ann. Acad. Sci. Fenn. Ser. A I Math. 18(2) (1993), 369–375.Google Scholar
Bergweiler, W. and Morosawa, S.. Semihyperbolic entire functions. Nonlinearity 15(5) (2002), 1673–1684.CrossRefGoogle Scholar
Beardon, A. F. and Minda, D.. The hyperbolic metric and geometric function theory. Quasiconformal Mappings and their Applications. Narosa, New Delhi, 2007.Google Scholar
Bonk, M. and Meyer, D.. Expanding Thurston Maps (Mathematical Surveys and Monographs, 225). American Mathematical Society, Providence, RI, 2017.CrossRefGoogle Scholar
Bolsch, A.. Periodic Fatou components of meromorphic functions. Bull. Lond. Math. Soc. 31(5) (1999), 543–555.CrossRefGoogle Scholar
Douady, A. and Hubbard, J. H. Étude dynamique des polynomes complexes. Université de Paris-Sud, Département de Mathématique, Orsay, 1984.Google Scholar
Erëmenko, A. and Lyubich, M.. Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42(4) (1992), 989–1020.CrossRefGoogle Scholar
Greenberg, M. and Harper, J.. Algebraic Topology. A First Course (Mathematics Lecture Note Series, 58). Benjamin Cummings Publishing Company, Reading, MA, 1981.Google Scholar
Graczyk, J. and Światek, G.. The Real Fatou Conjecture (Annals of Mathematics Studies, 144). Princeton University Press, Princeton, NJ, 1998.CrossRefGoogle Scholar
Hatcher, A.. Algebraic Topology. Cambridge University Press, Cambridge, 2002.Google Scholar
Heins, M.. Asymptotic spots of entire and meromorphic functions. Ann. of Math. 66(3) (1957), 430–439.CrossRefGoogle Scholar
Herring, M. E.. Mapping properties of Fatou components. Ann. Acad. Sci. Fenn. Math. 23(2) (1998), 263–274.Google Scholar
Hubbard, J. H.. Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 1: Teichmüller Theory. Matrix Editions, Ithaca, NY, 2006.Google Scholar
Lehto, O. and Virtanen, K.. Quasiconformal Mappings in the Plane (Grundlehren der mathematischen Wissenschaften, 126), 2nd edn. Springer, New York, 1973. Translated from the German by K. W. Lucas.CrossRefGoogle Scholar
McMullen, C.. Complex Dynamics and Renormalization (AM-135). Princeton University Press, Princeton, NJ, 1994.Google Scholar
Mihaljević-Brandt, H.. Topological dynamics of transcendental entire functions. PhD Thesis, University of Liverpool, 2009.Google Scholar
Mihaljević-Brandt, H.. A landing theorem for dynamic rays of geometrically finite entire functions. J. Lond. Math. Soc. 81(3) (2010), 696–714.CrossRefGoogle Scholar
Mihaljević-Brandt, H.. Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds. Trans. Amer. Math. Soc. 364(8) (2012), 4053–4083.CrossRefGoogle Scholar
Milnor, J.. Dynamics in One Complex Variable (AM-160) (Annals of Mathematics Studies, 160), 3rd edn. Princeton University Press, Princeton, NJ, 2006.Google Scholar
Mihaljević-Brandt, H. and Rempe-Gillen, L.. Absence of wandering domains for some real entire functions with bounded singular sets. Math. Ann. 357(4) (2013), 1577–1604.CrossRefGoogle Scholar
Munkres, J. R.. Topology, 2nd edn. Prentice Hall, Upper Saddle River, NJ, 2000.Google Scholar
Pardo-Simón, L.. Dynamics of transcendental entire functions with escaping singular orbits. PhD Thesis, University of Liverpool, 2019.Google Scholar
Pardo-Simón, L.. Splitting hairs with transcendental entire functions. Preprint, 2019, arXiv:1905.03778.Google Scholar
Pardo-Simón, L.. Topological dynamics of cosine maps. Preprint, 2020, arXiv:2003.07250.Google Scholar
Pommerenke, Ch.. Boundary Behaviour of Conformal Maps (Grundlehren der Mathematischen Wissenschaften, 299). Springer, Berlin, 1992.CrossRefGoogle Scholar
Rempe, L.. Rigidity of escaping dynamics for transcendental entire functions. Acta Math. 203(2) (2009), 235–267.CrossRefGoogle Scholar
Rempe-Gillen, L.. On prime ends and local conectivity. Preprint, 2014, arXiv:math/0309022v6 [math.GN].Google Scholar
Rottenfußer, G., Rückert, J., Rempe, L. and Schleicher, D.. Dynamic rays of bounded-type entire functions. Ann. of Math. (2) 173(1) (2011), 77–125.CrossRefGoogle Scholar
Rempe-Gillen, L. and Sixsmith, D. J.. Hyperbolic entire functions and the Eremenko–Lyubich class: class $\mathbf{\mathcal{B}}$ or not class $\mathbf{\mathcal{B}}$ ? Math. Z. 286(3–4) (2016), 783–800.CrossRefGoogle Scholar
Schleicher, D.. Dynamics of entire functions. Holomorphic Dynamical Systems (Lecture Notes in Mathematics, 1998). Springer, Heidelberg, 2010, pp. 295–339.CrossRefGoogle Scholar
Sixsmith, D. J.. Dynamical sets whose union with infinity is connected. Ergod. Th. & Dynam. Sys. 40 (2020), 789–798.CrossRefGoogle Scholar
Shen, Z. and Rempe-Gillen, L.. The exponential map is chaotic: an invitation to transcendental dynamics. Amer. Math. Monthly 122(10) (2015), 919–940.CrossRefGoogle Scholar
Thurston, W. P.. The geometry and topology of three-manifolds. Lecture Notes, 1984.Google Scholar
Thurston, W. P.. On the combinatorics and dynamics of iterated rational maps. Preprint, 1984.Google Scholar
Torhorst, M.. Über den Rand der einfach zusammenhängenden ebenen Gebiete. Math. Z. 9(1–2) (1921), 44–65.CrossRefGoogle Scholar
Vuorinen, M.. Conformal Geometry and Quasiregular Mappings (Lecture Notes in Mathematics, 1319). Springer, Berlin, 1988.CrossRefGoogle Scholar