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

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.