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.
Citation
Hasina Akter. Mariusz Urbański. "Real analyticity of Hausdorff dimension of Julia sets of parabolic polynomials $f_{\lambda}(z)=z(1-z-\lambda z^{2})$." Illinois J. Math. 55 (1) 157 - 184, Spring 2011. https://doi.org/10.1215/ijm/1355927032
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