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Non-normality, topological transitivity and expanding families

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  14 December 2021

THIERRY MEYRATH  and
JÜRGEN MÜLLER
THIERRY MEYRATH
Affiliation:
University of Luxembourg, Department of Mathematics, L-4364 Esch-sur-Alzette, Luxembourg. e-mail: thierry.meyrath@uni.lu
JÜRGEN MÜLLER
Affiliation:
Universität Trier, Fachbereich IV – Mathematik, D-54286 Trier, Germany. e-mail: jmueller@uni-trier.de
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Abstract

We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel’s Theorem. In particular, we also obtain characterisations of non-normality in terms of such properties.

MSC classification

Primary: 30D45: Bloch functions, normal functions, normal families 30D30: Meromorphic functions, general theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 3 , November 2022 , pp. 511 - 523
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Beise, H.-P., Meyrath, T. and Müller, J.. Universality properties of Taylor series inside the domain of holomorphy. J. Math. Anal. Appl. 383 (2011), 234–238.CrossRefGoogle Scholar
Beise, H.-P., Meyrath, T. and Müller, J.. Limit functions of discrete dynamical systems. Conform. Geom. Dyn. 18 (2014), 56–64.CrossRefGoogle Scholar
Beise, H.-P., Meyrath, T. and Müller, J.. Mixing Taylor shifts and universal Taylor series. Bull. London Math. Soc. 47 (2015), 136–142.CrossRefGoogle Scholar
Bernal-gonzález, L., Jung, A. and Müller, J.. Universality vs. non-normality of families of meromorphic functions. Proc. Amer. Math. Soc. 149 (2021), 761–771.CrossRefGoogle Scholar
Blatt, H. P., Blatt, S. and Luh, W.. On a generalization of Jentzsch’s theorem. J. Approx. Theory 159 (2009), 26–38.CrossRefGoogle Scholar
Carleson, L. and Gamelin, T. W.. Complex Dynamics (Springer, New York, 1993).Google Scholar
Chen, J.-F.. Exceptional functions and normal families of holomorphic functions with multiple zeros. Georgian Math. J. 18 (2011), 31–38.CrossRefGoogle Scholar
Chen, Q., Pang, X. and Yang, P.. A new Picard type theorem concerning elliptic functions. Ann. Acad. Sci. Fenn. Math. 40 (2015), 17–30.CrossRefGoogle Scholar
Cheng, C. and Xu, Y.. Normality concerning exceptional functions. Rocky Mt. J. Math. 45 (2015), 157–168.CrossRefGoogle Scholar
Chuang, C.-T.. Normal families of meromorphic functions, (World Scientific, 1993).CrossRefGoogle Scholar
Clunie, J. and Hayman, W. K.. The spherical derivative of integral and meromorphic functions. Comment. Math. Helv. 40 (1966), 117–148.CrossRefGoogle Scholar
Conway, J. B.. Functions of One Complex Variable, I 2nd ed., (Springer, New York, 1978).CrossRefGoogle Scholar
Dvoretzky, A.. On sections of power series. Ann. of Math. 51 (1950), 643–696.CrossRefGoogle Scholar
Fatou, P.. Sur l’itération des fonctions transcendantes entières. Acta Math. 47 (1926), 337–360.CrossRefGoogle Scholar
Gardiner, S.. Existence of universal Taylor series for nonsimply connected domains. Constr. Approx. 35 (2012), 245–257.CrossRefGoogle Scholar
Gehlen, W.. Overconvergent power series and conformal maps. J. Math. Anal. Appl. 198 (1996), 490–505.CrossRefGoogle Scholar
Gu, Y. X.. A normal criterion of meromorphic families. Sci. Sinica 1 (1979), 267–274.Google Scholar
Ivanov, K. G., Saff, E. B. and Totik, V.. On the behavior of zeros of polynomials of best and near-best approximation. Canad. J. Math. 43 (1991), 1010–1021.CrossRefGoogle Scholar
Jentzsch, R.. Untersuchungen zur Theorie der Folgen analytischer Funktionen. Acta. Math. 41 (1918) 219–251.CrossRefGoogle Scholar
Jentzsch, R.. Fortgesetzte Untersuchungen über die Abschnitte von Potenzreihen. Acta. Math. 41 (1918), 253–270.CrossRefGoogle Scholar
Grosse-erdmann, K.-G. and Peris, A.. Linear Chaos, (Springer, London, 2011).CrossRefGoogle Scholar
Kalmes, T., Müller, J. and Niess, M.. On the behaviour of power series in the absence of Hadamard–Ostrowski gaps. C. R. Math. Acad. Sci. Paris 351 (2013), 255–259.CrossRefGoogle Scholar
Melas, A.. Universal functions on nonsimply connected domains. Ann. Inst. Fourier (Grenoble) 51 (2001), 1539–1551.CrossRefGoogle Scholar
Minda, D.. Yosida functions, Lectures on Complex Analysis (Xian, 1987), Chuang, C.-T. (ed.) (World Scientific, Singapore, 1988), 197–213.Google Scholar
Misiurewicz, M.. On iterates of $e^z$ . Ergod. Theory Dynam. Systems. 1 (1981), 103–106.CrossRefGoogle Scholar
Morosawa, S., Nishimura, Y., Taniguchi, M. and Ueda, T.. Holomorphic Dynamics. (Cambridge University Press, Cambridge, 2000).Google Scholar
Nevo, S., Pang, X. and Zalcman, L.. Quasinormality and meromorphic functions with multiple zeros. J. Anal. Math. 101 (2007), 1–23.CrossRefGoogle Scholar
Saff, E. B. and Stahl, H.. Ray sequences of best rational approximants for $\left|{x}\right|^{\alpha}$ . Canad. J. Math. 49 (1997), 1034–1065.CrossRefGoogle Scholar
Schiff, J. L.. Normal Families, Springer, (New York, Berlin, Heidelberg, 1993).Google Scholar
Schleicher, D.. Dynamics of entire functions. In: Holomorphic Dynamical Systems. Lecture Notes in Math. vol. 1998 (Springer, Berlin, 2010), 295–339.Google Scholar
Wang, Y. and Fang, M.. Picard values and normal families of meromorphic functions with multiple zeros. Acta Math. Sinica (N.S.) 14 (1998), 17–26.Google Scholar
Yosida, K.. On a class of meromorphic functions. Proc. Phys.-Math. Soc. Japan 16 (1934), 227–235.Google Scholar