PERTURBATIONS IN THE SPEISER CLASS

In this paper we study perturbations of maps from a family of expanding entire functions from the Speiser class. Maps in this family, which we denoted by H, have the form ${f_a}\left( z \right) = \sum\nolimits_{j = 0}^n {{a_j}{e^{\left( {j - k} \right)z}}} $ where 0 < k < n and a = (a0, . . ....

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Bibliographic Details
Published in:The Rocky Mountain journal of mathematics 2007-01, Vol.37 (3), p.763-800
Main Authors: COICULESCU, ION, SKORULSKI, BARTŁOMIEJ
Format: Article
Language:English
Subjects:
28D10
37A05
37D35
37F10
37F35
37F45
Eigenvalues
Entire functions
Ergodic theory
Hausdorff dimensions
Homeomorphism
Infinity
Julia sets
Mathematical functions
Mathematical theorems
Mathematics
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Summary:In this paper we study perturbations of maps from a family of expanding entire functions from the Speiser class. Maps in this family, which we denoted by H, have the form ${f_a}\left( z \right) = \sum\nolimits_{j = 0}^n {{a_j}{e^{\left( {j - k} \right)z}}} $ where 0 < k < n and a = (a0, . . . , an) ε Cn+1 is a parameter. Using a known result of Eremenko and Lyubich about structural stability of such maps, perturbation theory (Kato-Rellich theorem) and research of Urbański and Zdunik on perturbations in the exponential family, we shall prove that the Hausdorff dimension of the set of points in the Julia set having nonescaping orbits depends analytically on the parameter a ε Cn+1.
ISSN:0035-7596
1945-3795
DOI:10.1216/rmjm/1182536163