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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: [email protected])
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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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