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Hausdorff dimension of the set of elliptic functions with critical values approaching infinity

Published online by Cambridge University Press:  01 October 2012

PIOTR GAŁĄZKA
PIOTR GAŁĄZKA*
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw 00-661, Poland. e-mail: P.Galazka@mini.pw.edu.pl
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Abstract

Let Λ denote the Weierstrass function with a period lattice Λ. We consider escaping parameters in the family βΛ, i.e. the parameters β for which the orbits of all critical values of βΛ approach infinity under iteration. Unlike the exponential family, the functions considered here are ergodic and admit a non-atomic, σ-finite, ergodic, conservative and invariant measure μ absolutely continuous with respect to the Lebesgue measure. Under additional assumptions on Λ, we estimate the Hausdorff dimension of the set of escaping parameters in the family βΛ from below, and compare it with the Hausdorff dimension of the escaping set in the dynamical space, proving a similarity between the parameter plane and the dynamical space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 1 , January 2013 , pp. 97 - 118
Copyright
Copyright © Cambridge Philosophical Society 2012

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