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Hausdorff Dimension of Sets of Escaping Points and Escaping Parameters for Elliptic Functions

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

Published online by Cambridge University Press:  30 December 2015

Piotr Gałązka  and
Janina Kotus
Piotr Gałązka
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland (p.galazka@mini.pw.edu.pl; j.kotus@impan.pl)
Janina Kotus
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland (p.galazka@mini.pw.edu.pl; j.kotus@impan.pl)
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Abstract

Let be a non-constant elliptic function. We prove that the Hausdorff dimension of the escaping set of f equals 2q/(q+1), where q is the maximal multiplicity of poles of f. We also consider the escaping parameters in the family fβ = βf, i.e. the parameters β for which the orbit of one critical value of fβ escapes to infinity. Under additional assumptions on f we prove that the Hausdorff dimension of the set of escaping parameters ε in the family fβ is greater than or equal to the Hausdorff dimension of the escaping set in the dynamical space. This demonstrates an analogy between the dynamical plane and the parameter plane in the class of transcendental meromorphic functions.

Keywords

meromorphic functionsJulia setcritical valuesescaping setescaping parametersHausdorff dimension

MSC classification

Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 671 - 690
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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