Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials

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Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have ... continued below

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Akter, Hasina August 2012.

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Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed Markov systems with infinite number of edges in order to reduce the problem of real analyticity of Hausdorff dimension for the given family of polynomials to prove the corresponding statement for the family .

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  • August 2012

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  • Feb. 1, 2014, 6:14 p.m.

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  • March 14, 2019, 10:42 a.m.

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Akter, Hasina. Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials, dissertation, August 2012; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc271768/: accessed May 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; .