Quasisymmetric rigidity, carpet Julia sets and the landing of dynamic resp. parameter rays

Abstract : The thesis consists of five parts. The first part is concerned with the quasisymmetric rigidity of a new Sierpinski carpet, which are not quasis-ymmetrically equivalent to the standard Sierpinski carpets. The second part discusses the quasisymmetrically geometry of the carpet Julia sets, including the uniformly quasicircle and uniformly separated properties. The third part is to determine when two external rays of a polynomial land at the same point. As an application, we also show the monotonicity of core-entropy on a family of quadratic polynomials. In the fourth part, following Cui and Tan's work, we use the classic tools modulus of annulus and quasi-conformal surgery to study the landing of some parameter rays in shift locus parameter space. The last part discusses a family of generated renormal-ization transformations. Specifically, it is on the connec-tivity of its Julia sets and the non-escaping locus in its parameter space, the asymptotic formula of the Hausdorff dimention of the Julia sets.
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Jinsong Zeng. Quasisymmetric rigidity, carpet Julia sets and the landing of dynamic resp. parameter rays. Data Structures and Algorithms [cs.DS]. Université d'Angers; Université de Fudan (Shanghai, Chine), 2015. English. ⟨NNT : 2015ANGE0054⟩. ⟨tel-02131495⟩

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