Holomorphic Dynamics: On Accumulation of Stretching Rays

Pia B.N. Willumsen

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    Abstract

    In this thesis we study the structure of the boundary of the cubic connectedness locus viewed from the escape locus; i.e. the limiting behaviour of stretching rays.We prove that the stretching ray through a polynomial P with no parabolic fixed point of multiplier one accumulates a cubic polynomial with one parabolic fixed point of multiplier one, the other fixed point repelling, and each critical point in the parabolic basin or in a (super)-attracting basin only if a critical point of P belongs to a fixed ray of the filled Julia set of P. This applies to the locus of cubic polynomials with both critical points in the immediate basin of parabolic fixed points of multiplier one.By using a holomorphic index argument we prove that there are regions in the boundary of the cubic connectedness locus where no strtching rays accumulate.Finally we introduce the concept of a ground wind at polynomials with a parabolic cycle. We give an example of a non-vanishing cubic ground wind. It follows that the wring operator is discontinuous in degree three.
    Original languageEnglish
    Place of PublicationLyngby
    PublisherDepartment of Mathematics, Technical University of Denmark
    Number of pages97
    Publication statusPublished - 1997

    Cite this

    Willumsen, P. B. N. (1997). Holomorphic Dynamics: On Accumulation of Stretching Rays. Department of Mathematics, Technical University of Denmark.