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The M-set of λ exp(z)/z has infinite area

Part of: Complex dynamical systems Holomorphic mappings and correspondences Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  11 January 2016

Guoping Zhan  and
Liangwen Liao
Guoping Zhan
Affiliation:
Department of Mathematics, Zhejiang University of Technology, Hangzhou 310023, People's Republic of China, zhangp@zjut.edu.cn
Liangwen Liao
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China, maliao@nju.edu.cn
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Abstract

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It is known that the Fatou set of the map exp(z)/z defined on the punctured plane ℂ* is empty. We consider the M-set of λ exp(z)/z consisting of all parameters λ for which the Fatou set of λexp(z)/z is empty. We prove that the M-set of λexp(z)/z has infinite area. In particular, the Hausdorff dimension of the M-set is 2. We also discuss the area of complement of the M-set.

MSC classification

Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations 32H50: Iteration problems 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 217 , March 2015 , pp. 133 - 159
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Baker, I. N., Kotus, J., and Lü, Y., Iterates of meromorphic functions, IV: Critically finite functions, Results Math. 22 (1992), 651–656. MR 1189754. DOI 10.1007/BF03323112.Google Scholar
[2] Bergweiler, W., Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), 151–188. MR 1216719. DOI 10.1090/S0273-0979-1993-00432-4.CrossRefGoogle Scholar
[3] Carleson, L. and Gamelin, T. W., Complex Dynamics, Universitext, Springer, New York, 1993. MR 1230383. DOI 10.1007/978-1-4612-4364-9.Google Scholar
[4] Devaney, R., Julia sets and bifurcation diagrams for exponential maps, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 167–171. MR 0741732. DOI 10.1090/S0273-0979-1984-15253-4.Google Scholar
[5] Devaney, R., Fagella, N., and Jarque, X., Hyperbolic components of the complex exponential family, Fund. Math. 174 (2002), 193–215. MR 1924998. DOI 10.4064/fm174-3-1.Google Scholar
[6] Devaney, R. and Goldberg, L. R., Uniformization of attracting basins for exponential maps, Duke Math. J. 55 (1987), 253–266. MR 0894579. DOI 10.1215/S0012-7094-87-05513-X.Google Scholar
[7] Devaney, R., Goldberg, L., and Hubbard, J. H., A dynamical approximation to the exponential map by polynomials, preprint, 1985.Google Scholar
[8] Lyubich, M. Y., Measurable dynamics of the exponential (in Russian), Sibirsk. Mat. Zh. 28, no. 5 (1987), 111–127; English translation in Siberian Math. J. 28, no. 5 (1987), 780–797. MR 0924986.Google Scholar
[9] Makienko, P. M., Iterations of analytic functions in ℂ* (in Russian), Dokl. Akad. Nauk 297, no. 1 (1987), 35–37; English translation in Dokl. Math. 36, no. 3 (1988), 418-420. MR 0916928.Google Scholar
[10] McMullen, C., Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300, no. 1 (1987), 329–342. MR 0871679. DOI 10.2307/2000602.Google Scholar
[11] Misiurewicz, M., On iterates of ez , Ergodic Theory Dynam. Systems 1 (1981), 103–106. MR 0627790.Google Scholar
[12] Qiu, W., Hausdorff dimension of the M-set of λexp(z), Acta Math. Sinica (N.S.) 10 (1994), 362–368. MR 1416147. DOI 10.1007/BF02582032.Google Scholar
[13] Rempe, L., Topological dynamics of exponential maps on their escaping sets, Ergodic Theory Dynam. Systems 26 (2006), 1939–1975. MR 2279273. DOI 10.1017/S0143385706000435.Google Scholar
[14] Rempe, L., The escaping set of the exponential, Ergodic Theory Dynam. Systems 30 (2010), 595–599. MR 2599894. DOI 10.1017/S014338570900008X.CrossRefGoogle Scholar
[15] Rempe, L., Dynamics of exponential maps, Ph.D. dissertation, Christian-Albrechts-Universität zu Kiel, Kiel, Germany, 2003.Google Scholar
[16] Rempe, L. and Schleicher, D., “Combinatorics of bifurcations in exponential parameter space” in Transcendental Dynamics and Complex Analysis, London Math. Soc. Lecture Note Ser. 348, Cambridge University Press, Cambridge, 2008, 317–370. MR 2458808. DOI 10.1017/CBO9780511735233.014.Google Scholar
[17] Rempe, L. and Schleicher, D., Bifurcations in the space of exponential maps, Invent. Math. 175 (2009), 103–135. MR 2461427. DOI 10.1007/s00222-008-0147-5.Google Scholar
[18] Wang, X. and Zhang, G., Most of the maps near the exponential are hyperbolic, Nagoya Math. J. 191 (2008), 135–148. MR 2451223.Google Scholar
[19] Zhan, G. P., Non-recurrence of exp(z)/z , Acta Math. Sin. (Engl. Ser.) 29 (2013), 703–716. MR 3029285. DOI 10.1007/s10114-012-1451-y.Google Scholar
[20] Zhan, G. P., Hausdorff dimension of a fractal set of exp(z)/z, preprint, 2012.Google Scholar
[21] Zhan, G. P. and Liao, L.-W., Area of non-escaping parameters of the sine family, Houston J. Math. 38 (2012), 493–524. MR 2954649.Google Scholar