We use a new type of distortion control of univalent functions to give an alternative proof of Douady-Hubbard’s ray-landing theorem for quadratic Misiurewicz polynomials. The univalent maps arise from Thurston’s iterated algorithm on perturbation of such polynomials.
CUI GuiZhen 1, & TAN Lei 2 1 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
We show that the Hausdorff dimension of quasi-circles of polygonal mappings is one. Furthermore, we apply this result to the theory of extremal quasiconformal mappings. Let [μ] be a point in the universal Teichmiiller space such that the Hausdorff dimension of fμ（δ△） is bigger than one. We show that for every kn ∈ （0, 1） and polygonal differentials δn, n = 1, 2, the sequence {[kn δn/｜δn｜} cannot converge to [μ] under the Teichmiiller metric.
In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Fuhermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpirlski curve is given.
The convergence of linear fractional transformations is an important topic in mathematics.We study the pointwise convergence of p-adic Mbius maps,and classify the possibilities of limits of pointwise convergent sequences of Mbius maps acting on the projective line P1(C p),where C p is the completion of the algebraic closure of Q p.We show that if the set of pointwise convergence of a sequence of p-adic Mbius maps contains at least three points,the sequence of p-adic Mbius maps either converges to a p-adic Mbius map on the projective line P1(C p),or converges to a constant on the set of pointwise convergence with one unique exceptional point.This result generalizes the result of Piranian and Thron(1957)to the non-archimedean settings.
