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We prove that D0, the set of all parameters λ ∈ C \ {0} for which the cubic polynomial fλ is parabolic and has no other parabolic or finite attracting periodic cycles, contains a deleted neighborhood of the origin 0. Our main result is that the function D0 3 λ 7→ HD(J(fλ)) ∈ R is real-analytic. This function ascribes to the polynomial fλ the Hausdorff dimension of its Julia set J(fλ). The theory of parabolic and hyperbolic graph directed Markov systems with infinite number of edges is used in the proofs.

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