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Green's function and anti-holomorphic dynamics on a torus

Authors: Walter Bergweiler and Alexandre Eremenko
Journal: Proc. Amer. Math. Soc. 144 (2016), 2911-2922
MSC (2010): Primary 31A05, 33E05, 37F10
Published electronically: March 16, 2016
MathSciNet review: 3487224
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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.

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  • [1] Lars V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. MR 0054016 (14,857a)
  • [2] Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743 (50 #10211)
  • [3] 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 (91k:33016)
  • [4] 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 (2011c:30108)
  • [5] Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151-188. MR 1216719 (94c:30033),
  • [6] 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 (96h:30055),
  • [7] 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 (2011f:85001),
  • [8] 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 (2006d:30038),
  • [9] 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
  • [10] Ching-Li Chai, Chang-Shou Lin and Chin-Lung Wang, Mean field equations, hyperelliptic curves and modular forms: II, arXiv: 1502.03295.
  • [11] 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 (91c:58072)
  • [12] A. Eremenko, Distribution of zeros of some real polynomials and iteration theory, preprint, Inst. for Low Temperature Physics and Engineering, Kharkov, 1989.
  • [13] P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 33-94 (French). MR 1504792
  • [14] Steven R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Cambridge, 2003. MR 2003519 (2004i:00001)
  • [15] G. H. Halphen, Traité des fonctions elliptiques et de leurs applications, Gauthier-Villars, Paris, 1886.
  • [16] 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 (94f:30035),
  • [17] A. Hurwitz, Sur les points critiques des fonctions inverses, C. R. Acad. Sci. Paris 144 (1907) 63-65.
  • [18] Adolf Hurwitz, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie von R. Courant. Mit einem Anhang von H. Röhrl. Vierte vermehrte und verbesserte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 3, Springer-Verlag, Berlin-New York, 1964 (German). MR 0173749 (30 #3959)
  • [19] 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 (2012a:30070),
  • [20] 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 (electronic). MR 2196041 (2007h:26018),
  • [21] 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 (2009f:30003)
  • [22] Dmitry Khavinson and Grzegorz Światek, On the number of zeros of certain harmonic polynomials, Proc. Amer. Math. Soc. 131 (2003), no. 2, 409-414. MR 1933331 (2003j:30015),
  • [23] S.-Y. Lee and N. G. Makarov, Topology of quadrature domains, J. Amer. Math. Soc., electronically published on May 11, 2015, DOI: (to appear in print).
  • [24] 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 (2012a:35111),
  • [25] Curt McMullen, Families of rational maps and iterative root-finding algorithms, Ann. of Math. (2) 125 (1987), no. 3, 467-493. MR 890160 (88i:58082),
  • [26] John W. Milnor, Topology from the differentiable viewpoint, Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va., 1965. MR 0226651 (37 #2239)
  • [27] John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309 (2006g:37070)
  • [28] Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330 (15,208c)
  • [29] 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: (to appear in print).
  • [30] 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 (2004i:37099),
  • [31] 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 (2010m:37076),
  • [32] K. Weierstrass, Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen, edited by H. A. Schwarz, Springer, Berlin, 1893.
  • [33] 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 (97k:01072)

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Additional Information

Walter Bergweiler
Affiliation: Mathematisches Seminar, Christian–Albrecht–Universität zu Kiel, Ludewig–Meyn–Straße 4, 24098 Kiel, Germany

Alexandre Eremenko
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

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
Article copyright: © Copyright 2016 American Mathematical Society

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