Abstract: Let $D_0$ denote the set of all parameters $\lambda\in\C\setminus\{0\}$ for which the cubic polynomial $f_\lambda$ is parabolic and has no parabolic or finite attracting periodic cycles other than $0$. We prove that $D_0$ contains a deleted neighborhood of the origin $0$. Our main result is that the function $D_0\ni\lambda\mapsto\text{HD}(J(f_\lambda))\in\R$ 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.

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