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escaping parameters

Proceedings of the Edinburgh Mathematical Society (1)

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1 results

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
Piotr Gałązka, Janina Kotus
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 59 / Issue 3 / August 2016

Published online by Cambridge University Press:

30 December 2015, pp. 671-690
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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.