Green’s function and anti-holomorphic dynamics on a torus
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- by Walter Bergweiler and Alexandre Eremenko PDF
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Abstract:
We give a new, simple proof of the fact recently discovered by C.-S. Lin and C.-L. Wang that the Green function of a torus has either three or five critical points, depending on the modulus of the torus. The proof uses anti-holomorphic dynamics. As a byproduct we find a one-parametric family of anti-holomorphic dynamical systems for which the parameter space consists only of hyperbolic components and analytic curves separating them.References
- Lars V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0054016
- Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
- N. I. Akhiezer, Elements of the theory of elliptic functions, Translations of Mathematical Monographs, vol. 79, American Mathematical Society, Providence, RI, 1990. Translated from the second Russian edition by H. H. McFaden. MR 1054205, DOI 10.1090/mmono/079
- A. F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Quasiconformal mappings and their applications, Narosa, New Delhi, 2007, pp. 9–56. MR 2492498
- Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188. MR 1216719, DOI 10.1090/S0273-0979-1993-00432-4
- Walter Bergweiler and Alexandre Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), no. 2, 355–373. MR 1344897, DOI 10.4171/RMI/176
- Walter Bergweiler and Alexandre Eremenko, On the number of solutions of a transcendental equation arising in the theory of gravitational lensing, Comput. Methods Funct. Theory 10 (2010), no. 1, 303–324. MR 2676458, DOI 10.1007/BF03321770
- Walter Bergweiler, Alex Eremenko, and Jim K. Langley, Zeros of differential polynomials in real meromorphic functions, Proc. Edinb. Math. Soc. (2) 48 (2005), no. 2, 279–293. MR 2157248, DOI 10.1017/S0013091504000690
- Ching-Li Chai, Chang-Shou Lin, and Chin-Lung Wang, Mean field equations, hyperelliptic curves and modular forms: I, Camb. J. Math. 3 (2015), no. 1-2, 127–274. MR 3356357, DOI 10.4310/CJM.2015.v3.n1.a3
- Ching-Li Chai, Chang-Shou Lin and Chin-Lung Wang, Mean field equations, hyperelliptic curves and modular forms: II, arXiv: 1502.03295.
- W. D. Crowe, R. Hasson, P. J. Rippon, and P. E. D. Strain-Clark, On the structure of the Mandelbar set, Nonlinearity 2 (1989), no. 4, 541–553. MR 1020441, DOI 10.1088/0951-7715/2/4/003
- A. Eremenko, Distribution of zeros of some real polynomials and iteration theory, preprint, Inst. for Low Temperature Physics and Engineering, Kharkov, 1989.
- P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 33–94 (French). MR 1504792, DOI 10.24033/bsmf.1003
- Steven R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Cambridge, 2003. MR 2003519, DOI 10.1017/CBO9780511550447
- G. H. Halphen, Traité des fonctions elliptiques et de leurs applications, Gauthier-Villars, Paris, 1886.
- A. Hinkkanen, Iteration and the zeros of the second derivative of a meromorphic function, Proc. London Math. Soc. (3) 65 (1992), no. 3, 629–650. MR 1182104, DOI 10.1112/plms/s3-65.3.629
- A. Hurwitz, Sur les points critiques des fonctions inverses, C. R. Acad. Sci. Paris 144 (1907) 63–65.
- Adolf Hurwitz, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Die Grundlehren der mathematischen Wissenschaften, Band 3, Springer-Verlag, Berlin-New York, 1964 (German). Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie von R. Courant. Mit einem Anhang von H. Röhrl; Vierte vermehrte und verbesserte Auflage. MR 0173749
- Dmitry Khavinson and Erik Lundberg, Transcendental harmonic mappings and gravitational lensing by isothermal galaxies, Complex Anal. Oper. Theory 4 (2010), no. 3, 515–524. MR 2719790, DOI 10.1007/s11785-010-0050-0
- Dmitry Khavinson and Genevra Neumann, On the number of zeros of certain rational harmonic functions, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1077–1085. MR 2196041, DOI 10.1090/S0002-9939-05-08058-5
- Dmitry Khavinson and Genevra Neumann, From the fundamental theorem of algebra to astrophysics: a “harmonious” path, Notices Amer. Math. Soc. 55 (2008), no. 6, 666–675. MR 2431564
- Dmitry Khavinson and Grzegorz Świa̧tek, On the number of zeros of certain harmonic polynomials, Proc. Amer. Math. Soc. 131 (2003), no. 2, 409–414. MR 1933331, DOI 10.1090/S0002-9939-02-06476-6
- S.-Y. Lee and N. G. Makarov, Topology of quadrature domains, J. Amer. Math. Soc., electronically published on May 11, 2015, DOI: http://dx.doi.org/10.1090/jams828 (to appear in print).
- Chang-Shou Lin and Chin-Lung Wang, Elliptic functions, Green functions and the mean field equations on tori, Ann. of Math. (2) 172 (2010), no. 2, 911–954. MR 2680484, DOI 10.4007/annals.2010.172.911
- Curt McMullen, Families of rational maps and iterative root-finding algorithms, Ann. of Math. (2) 125 (1987), no. 3, 467–493. MR 890160, DOI 10.2307/1971408
- John W. Milnor, Topology from the differentiable viewpoint, University Press of Virginia, Charlottesville, Va., 1965. Based on notes by David W. Weaver. MR 0226651
- John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
- Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330, DOI 10.1007/978-3-662-06842-7
- Sabyasachi Mukherjee, Shizuo Nakane, and Dierk Schleicher, On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials, Ergodic Theory Dynam. Systems, electronically published on November 27, 2015, DOI: http://dx.doi.org/10.1017/etds.2015.65 (to appear in print).
- Shizuo Nakane and Dierk Schleicher, On multicorns and unicorns. I. Antiholomorphic dynamics, hyperbolic components and real cubic polynomials, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 10, 2825–2844. MR 2020986, DOI 10.1142/S0218127403008259
- William P. Thurston, On the geometry and dynamics of iterated rational maps, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher. MR 2508255, DOI 10.1201/b10617-3
- K. Weierstrass, Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen, edited by H. A. Schwarz, Springer, Berlin, 1893.
- E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
Additional Information
- Walter Bergweiler
- Affiliation: Mathematisches Seminar, Christian–Albrecht–Universität zu Kiel, Ludewig–Meyn–Straße 4, 24098 Kiel, Germany
- MR Author ID: 35350
- Email: bergweiler@math.uni-kiel.de
- Alexandre Eremenko
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 63860
- Email: eremenko@math.purdue.edu
- Received by editor(s): July 7, 2015
- Published electronically: March 16, 2016
- Additional Notes: The second author was supported by NSF grant DMS-1361836.
- Communicated by: Svitlana Mayboroda
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2911-2922
- MSC (2010): Primary 31A05, 33E05, 37F10
- DOI: https://doi.org/10.1090/proc/13044
- MathSciNet review: 3487224