### 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 language | English |
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Place of Publication | Lyngby |
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Publisher | Department of Mathematics, Technical University of Denmark |

Number of pages | 97 |

Publication status | Published - 1997 |

## Cite this

Willumsen, P. B. N. (1997).

*Holomorphic Dynamics: On Accumulation of Stretching Rays*. Department of Mathematics, Technical University of Denmark.