### On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials ${P}_{c}\left(z\right)={z}^{d}+c$ for c ∈ ℂ. Denote by ${J}_{c}$ the Julia set of ${P}_{c}$ and let ${\mathcal{M}}_{d}=c|{J}_{c}isconnected$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c\u2080\in \partial {\mathcal{M}}_{d}$: those for which the critical point 0 is not recurrent by ${P}_{c\u2080}$ and without parabolic cycles. The Hausdorff dimension of ${J}_{c}$, denoted by $HD\left({J}_{c}\right)$, does not depend continuously on c at such $c\u2080\in \partial {\mathcal{M}}_{d}$; on the other hand the function $c\mapsto HD\left({J}_{c}\right)$ is analytic in $\u2102-{\mathcal{M}}_{d}$. Our first result asserts that there is still some...