On Accumulation of Stretching Rays - Cubic Polynomials with a Multiple Fixed Point

The project studies the
parameter space for
polynomials of degree k,
where k is at least 3.
More precisely, we study the
boundary
of the connectedness locus
(the area in the
parameter space, where the
corresponding polynomials have
a connected Julia set). A
stretching ray is a real
analytical curve in the
complement of the
connectedness locus. We are
interested in the limiting
behaviour of stretching rays
when approaching the
connectedness locus from the
outside. Already a
degree three polynomial with a
multiple fixpoint exhibits
interesting behaviour: An
area in the boundary of the
cubic connectedness locus
has been found, where no
stretching ray lands. The
corresponding cubic polynomials
are
parabolic-attracting with
both critical points
in the
immediate basin
of the multiple fixpoint.
We have established necessary
conditions for the stretching
ray through a polynomial to
land in this area. The work is
continued attempting to prove
that stretching rays can land
only on a specific graph in this
area and that only a
very special type of
stretching rays can land on
this graph. Keywords: Complex
dynamical systems, parabolic
cycles, stretching rays.