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Dynamical approximation and kernels of non-escaping hyperbolic components

Published online by Cambridge University Press:  14 June 2011

HELENA MIHALJEVIĆ-BRANDT
HELENA MIHALJEVIĆ-BRANDT*
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany (email: helenam@math.uni-kiel.de)
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Abstract

Let ℱn be families of entire functions, holomorphically parameterized by a complex manifold M. We consider those parameters in M that correspond to non-escaping hyperbolic functions, i.e. those maps f∈ℱn for which the postsingular set P(f) is a compact subset of the Fatou set ℱ(f) of f. We prove that if ℱn→ℱ∞ in the sense of a certain dynamically sensible metric, then every non-escaping hyperbolic component in the parameter space of ℱ∞ is a kernel of a sequence of non-escaping hyperbolic components in the parameter spaces of ℱn. Parameters belonging to such a kernel do not always correspond to hyperbolic functions in ℱ∞. Nevertheless, we show that these functions must be J-stable. Using quasiconformal equivalences, we are able to construct many examples of families to which our results can be applied.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1418 - 1434
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Bergweiler, W., Kotus, J. and Lü, Y.. Iterates of meromorphic functions III: preperiodic domains. Ergod. Th. & Dynam. Sys. 11 (1991), 603–618.Google Scholar
[2]Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29(2) (1993), 151–188, arXiv:math.DS/9310226.CrossRefGoogle Scholar
[3]Bergweiler, W., Haruta, M., Kriete, H., Meier, H. G. and Terglane, N.. On the limit functions of iterates in wandering domains. Ann. Acad. Sci. Fenn., Ser. A I, Math. 18(2) (1993), 369–375.Google Scholar
[4]Bodelón, C., Devaney, R. L., Hayes, M., Roberts, G., Goldberg, L. R. and Hubbard, J. H.. Dynamical convergence of polynomials to the exponential. J. Difference Equ. Appl. 6(3) (2000), 275–307.CrossRefGoogle Scholar
[5]Buff, X. and Chéritat, A.. Upper bound for the size of quadratic Siegel disks. Invent. Math. 156(1) (2004), 1–24.CrossRefGoogle Scholar
[6]Devaney, R. L., Goldberg, L. R. and Hubbard, J. H.. A dynamical approximation to the exponential map by polynomials. Preprint, Berkeley, CA, 1986.Google Scholar
[7]Eremenko, A. E.. On the iteration of entire functions. Dynamical Systems and Ergodic Theory (Proceedings of Banach Center Publications Warsaw, 23). 1989, pp. 339–345.Google Scholar
[8]Eremenko, A. E. and Lyubich, M. Y.. Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42(4) (1992), 989–1020.CrossRefGoogle Scholar
[9]Goldberg, L. R. and Keen, L.. A finiteness theorem for a dynamical class of entire functions. Ergod. Th. & Dynam. Sys. 6 (1986), 183–192.CrossRefGoogle Scholar
[10]Golusin, G. M.. Geometrische Funktionentheorie. VEB Deutscher Verlag Der Wissenschaften, Berlin, 1957.Google Scholar
[11]Hubbard, J. H.. Teichmüller Theory and Applications to Geometry, Topology and Dynamics, Volume I: Teichmüller Theory. Matrix Editions, Ithaca, New York, 2006.Google Scholar
[12]Kisaka, M.. Local uniform convergence and convergence of Julia sets. Nonlinearity 8 (1995), 273–281.CrossRefGoogle Scholar
[13]Kisaka, M. and Shishikura, M.. On multiply connected wandering domains of entire functions. Transcendental Dynamics and Complex Analysis (London Mathematical Society Lecture Notes Series, 348). Eds. Rippon, P. J. and Stallard, G. M.. Cambridge University Press, Cambridge, 2008, pp. 217–250.CrossRefGoogle Scholar
[14]Krauskopf, B. and Kriete, H.. Kernel convergence of hyperbolic components. Ergod. Th. & Dynam. Sys. 17 (1997), 1137–1146.CrossRefGoogle Scholar
[15]Krauskopf, B. and Kriete, H.. On the convergence of hyperbolic components in families of finite type. Mathematica Gottingensis—Schriftenreihe des Mathematischen Instituts der Universität Göttingen, 1995. Available at http://www.uni-math.gwdg.de/mathgoe/.Google Scholar
[16]Lehto, O.. Univalent Functions and Teichmüller Spaces (Graduate Texts in Mathematics, 109). Springer, New York, Berlin, Heidelberg, 1986.Google Scholar
[17]Levin, B. Ya.. Lectures on Entire Functions (Translations of Mathematical Monographs, 150). American Mathematical Society, Providence, RI, 1996.CrossRefGoogle Scholar
[18]Mañé, R., Sad, P. and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4) 16(2) (1983), 193–217.CrossRefGoogle Scholar
[19]Milnor, J.. Dynamics in One Complex Variable, 3rd edn(Annals of Mathematics Studies, 160). Princeton University Press, Princeton, NJ, 2006.Google Scholar
[20]Nevanlinna, R.. Eindeutige analytische Funktionen, 2nd edn. Springer, Berlin, Göttingen, Heidelberg, 1953.CrossRefGoogle Scholar
[21]Semmler, G.. Complete interpolating sequences, the discrete Muckenhoupt condition, and conformal mapping. Ann. Acad. Sci. Fenn. Math. 35 (2007), 23–46, arXiv:math.DS/0708.2787.CrossRefGoogle Scholar
[22]Titchmarsh, E. C.. The Theory of Functions. Oxford University Press, Oxford, 1932.Google Scholar