Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials
per
Akter, Hasina
Major Professor
per
ski, Mariusz Urba
per
Allaart, Pieter C.
per
Fishman, Lior
www.unt.edu
Denton, Texas
University of North Texas
2012-08
eng
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 .
Real analyticity
Hausdorff dimension function
parabolic polynomial
UNTETD
UNT
public
Akter, Hasina
copyright
Copyright is held by the author, unless otherwise noted. All rights Reserved.
text_etd
text
Department of Mathematics
Mathematics
Doctoral
Doctor of Philosophy
University of North Texas
disse
Metadata created on 2014-01-23
rjwilson
DC
ark:/67531/metadc271768
2014-02-01, 18:14:03
dalemneh
2019-03-14, 10:42:41
False