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Given d ≥ 2 consider the family of polynomials for c ∈ ℂ. Denote by the Julia set of and let be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters : those for which the critical point 0 is not recurrent by and without parabolic cycles. The Hausdorff dimension of , denoted by , does not depend continuously on c at such ; on the other hand the function is analytic in . Our first result asserts that there is still some continuity...
Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map . We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set is continuous at σ₀ as the function of the parameter if and only if . Since on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of on an open and dense subset of ∂₀.
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