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Slices of parameter space for meromorphic maps with two asymptotic values

Part of: Entire and meromorphic functions, and related topics Complex dynamical systems Riemann surfaces Holomorphic functions of several complex variables

Published online by Cambridge University Press:  18 October 2021

TAO CHEN Open the ORCID record for TAO CHEN [Opens in a new window] ,
YUNPING JIANG Open the ORCID record for YUNPING JIANG [Opens in a new window]  and
LINDA KEEN Open the ORCID record for LINDA KEEN [Opens in a new window]
TAO CHEN
Affiliation:
Department of Mathematics, Engineering and Computer Science, Laguardia Community College, CUNY, 31-10 Thomson Ave. Long Island City, NY 11101, USA (e-mail: tchen@lagcc.cuny.edu)
YUNPING JIANG*
Affiliation:
Department of Mathematics, Queens College of CUNY, Flushing, NY 11367, USA
LINDA KEEN
Affiliation:
Department of Mathematics, CUNY Graduate School, New York, NY 10016, USA (e-mail: LKeen@gc.cuny.edu, linda.keenbrezin@gmail.com)
*
yunping.jiang@qc.cuny.edu
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Abstract

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work by Goldberg and Keen [The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift. Invent. Math. 101(2) (1990), 335–372] on degree two rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted.

Keywords

slices of parameter spacemeromorphic functionsshift locusvirtual center parametertransversality

MSC classification

Primary: 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems 37F20: Combinatorics and topology 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 30F30: Differentials on Riemann surfaces 30D30: Meromorphic functions, general theory 32A20: Meromorphic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 1 , January 2023 , pp. 99 - 139
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Ahlfors, L. and Bers, L.. Riemann’s mapping theorem for variable metrics. Ann. of Math. 72(2) (1960), 385–404.CrossRefGoogle Scholar
Beardon, A. F.. Iteration of Rational Functions. Springer, New York, 1991.CrossRefGoogle Scholar
Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151–188.CrossRefGoogle Scholar
Birman, J. S.. The algebraic structure of surface mapping class groups. Discrete Groups and Automorphic Functions (Proc. Conf., Cambridge, 1975). Academic Press, London, 1977, pp. 163–198.Google Scholar
Branner, B., Fagella, N.. Quasiconformal Surgery in Holomorphic Dynamics. Cambridge University Press, Cambridge, 2014.CrossRefGoogle Scholar
Baker, I. N., Kotus, J. and Lü, Y.. Iterates of meromorphic functions II: examples of wandering domains. J. Lond. Math. Soc. 42(2) (1990), 267–278.CrossRefGoogle Scholar
Baker, I. N., Kotus, J. and Lü, Y.. Iterates of meromorphic functions I. Ergod. Th. & Dynam. Sys. 11 (1991), 241–248.CrossRefGoogle Scholar
Baker, I. N., Kotus, J. and Lü, Y.. Iterates of meromorphic functions III: preperiodic domains. Ergod. Th. & Dynam. Sys. 11 (1991), 603–618.CrossRefGoogle Scholar
Baker, I. N., Kotus, J. and Lü, Y.. Iterates of meromorphic functions IV: critically finite functions. Results Math. 22 (1991), 651–656.Google Scholar
Chen, T., Jiang, Y. and Keen, L.. Cycle doubling, merging, and renormalization in the tangent family. Conform. Geom. Dyn. 22 (2018), 271–314.CrossRefGoogle Scholar
Chen, T., Jiang, Y. and Keen, L.. Accessible boundary points in the shift locus of a family of meromorphic functions with two asymptotic values. Arnold Math. J. Lyubich (2021), to appear.Google Scholar
Chen, T. and Keen, L.. Slices of parameter spaces of generalized Nevanlinna functions. Discrete Contin. Dyn. Syst. 39(10) (2019), 5659–5681.CrossRefGoogle Scholar
Devaney, R. L., Fagella, N. and Jarque, X.. Hyperbolic components of the complex exponential family. Fund. Math. 174 (2002), 193–215.CrossRefGoogle Scholar
Douady, A. and Hubbard, J.. On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér. (4) 18 (1985), 287–343.CrossRefGoogle Scholar
Devaney, R. and Keen, L.. Dynamics of tangent. Dynamical Systems, Proceedings (College Park, MD, 1986–87) (Lecture Notes in Mathematics, 1342). Springer-Verlag, Heidelberg, 1988, pp. 105–111.Google Scholar
Devaney, R. and Keen, L.. Dynamics of meromorphic functions: functions with polynomial Schwarzian derivative. Ann. Sci. Éc. Norm. Supér. (4) 22 (1989), 55–79.CrossRefGoogle Scholar
Fagella, N. and Garijo, A.. The parameter planes of $\lambda {z}^m{e}^z$ for $m\ge 2$ . Commun. Math. Phys. 273(3) (2007), 755–783.CrossRefGoogle Scholar
Fagella, N. and Keen, L.. Stable components in the parameter plane of transcendental functions of finite type. J. Geom. Anal. 31(5) (2021), 4816–4855.CrossRefGoogle Scholar
Gardiner, F., Jiang, Y. and Wang, Z.. Holomorphic motions and related topics geometry of Riemann surfaces. London Math. Soc. Lecture Note Ser. 368 (2010), 166–193.Google Scholar
Goldberg, L. R. and Keen, L.. The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift. Invent. Math. 101(2) (1990), 335–372.CrossRefGoogle Scholar
Hille, E.. Ordinary Differential Equations in the Complex Domain. John Wiley and Sons, New York, 1976.Google Scholar
Jiang, Y.. Renormalization and Geometry in One-Dimensional and Complex Dynamics (Advanced Series in Nonlinear Dynamics, 10). World Scientific Publishing Co. Pte. Ltd., River Edge, NJ, 1996.CrossRefGoogle Scholar
Keen, L.. Complex and real dynamics for the family $\lambda \tan(z)$ . Complex Dynamics and Related fields (Kyoto, 2001). Surikaisekikenkyusho Kokyuroku 1269(2002), 93–102 (in Japanese).Google Scholar
Keen, L. and Kotus, J.. Dynamics of the family of $\lambda \tan z$ . Conform. Geom. Dyn. 1 (1997), 28–57.CrossRefGoogle Scholar
Levin, G., van Strien, S. and Shen, W.. Monotonicity of entropy and positively oriented transversality for families of interval maps . Preprint, 2016, arXiv:1611.10056v1.Google Scholar
Massey, W. S.. Algebraic Topology: An Introduction. Harcourt, Brace and World, Inc., New York, 1967.Google Scholar
McMullen, C. T. and Sullivan, D. P.. Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system. Adv. Math. 135(2) (1998), 351–395.CrossRefGoogle Scholar
Milnor, J.. Dynamics in One Complex Variable (Annals of Mathematics Studies, 160), 3rd edn. Princeton University Press, Princeton, NJ, 2006.Google Scholar
Milnor, J.. Geometry and dynamics of quadratic rational maps. Exp. Math. 2 (1993), 37–83.CrossRefGoogle Scholar
Moser, J.. Stable and Random Motions in Dynamical Systems. Princeton University Press, Princeton, NJ, 1973.Google Scholar
Mané, R., Sad, P. and Sullivan, D. P.. On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4) 16 (1983), 193–217.CrossRefGoogle Scholar
Nayak, T.. Herman rings of meromorphic maps with an omitted value. Proc. Amer. Math. Soc. 144(2) (2016), 587–597.CrossRefGoogle Scholar
Nevanlinna, R.. Analytic Functions (Die Grundlehren der mathematischen Wissenschaften, 162). Springer-Verlag, Berlin, 1970. Translated from the 2nd edition by Philip Emig.CrossRefGoogle Scholar
Rempe-Gillen, L.. Dynamics of exponential maps. PhD Thesis, Christian-Albrechts-Universität Kiel, 2003.Google Scholar
Rickman, S.. Removability theorems for quasiconformal mappings. Ann. Acad. Sci. Fenn. Math. 449 (1969), 1–8.Google Scholar
Rippon, P. J. and Stallard, G. M.. Iteration of a class of hyperbolic meromorphic functions. Proc. Amer. Math. Soc. 127(11) (1999), 3251–3258.Google Scholar
Schleicher, D.. Attracting dynamics of exponential maps. Ann. Acad. Sci. Fenn. Math. 28 (2003), 3–34.Google Scholar
Wittner, B.. On the bifurcation loci of rational maps of degree two. PhD Thesis, Cornell University, 1988, unpublished.Google Scholar
Zheng, J.-H.. Dynamics of hyperbolic meromorphic functions. Discrete Contin. Dyn. Syst. 35(5) (2015), 2273–2298.CrossRefGoogle Scholar