Illinois Journal of Mathematics

Real analyticity of Hausdorff dimension of Julia sets of parabolic polynomials $f_{\lambda}(z)=z(1-z-\lambda z^{2})$

Hasina Akter and Mariusz Urbański

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Abstract

We prove that $D_*$, the set of all parameters $\lambda\in\mathbb{C}\setminus\{0\}$ for which the cubic polynomial $f_\lambda$ is parabolic and has no other parabolic or finite attracting periodic cycles, contains a deleted neighborhood $D_0$ of the origin 0. Our main result is that if $D_0$ is sufficiently small then the function $D_0\ni\lambda\mapsto\operatorname{HD}(J(f_\lambda))\in\mathbb{R}$ is real-analytic. This function ascribes to the polynomial $f_\lambda$ the Hausdorff dimension of its Julia set $J(f_\lambda)$. The theory of parabolic and hyperbolic graph directed Markov systems with infinite number of edges is used in the proofs.

Article information

Source
Illinois J. Math., Volume 55, Number 1 (2011), 157-184.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1355927032

Digital Object Identifier
doi:10.1215/ijm/1355927032

Mathematical Reviews number (MathSciNet)
MR3006684

Zentralblatt MATH identifier
1291.30241

Subjects
Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]

Citation

Akter, Hasina; Urbański, Mariusz. Real analyticity of Hausdorff dimension of Julia sets of parabolic polynomials $f_{\lambda}(z)=z(1-z-\lambda z^{2})$. Illinois J. Math. 55 (2011), no. 1, 157--184. doi:10.1215/ijm/1355927032. https://projecteuclid.org/euclid.ijm/1355927032


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