Non-locally Connected Quadratic Julia Sets

Work is done with non-linear
dynamical systems, which
appears from iterating with a
complex, quadratic polynomial.
The main purpose is to
increase the understanding of
the dynamic, geometry and
topology for polynomials with
locally connected Julia set.
As an example, the behaviour
of certain external rays under
repeated parabolic
pertubations are investigated.
As the most important side
results we should mentioned
the achievement of the first
known examples of connected,
but not curve-connected Julia
sets, as well as non-robust,
renormalizeable (infinitely
many times) quadratic
polynomials. Present work is
considering the possibility
to relate the polynomial
classes found to the
so-called Diophantic
conditions in the theory of
numbers and the possibility
to find quadratic Julia sets
with positive measure.
Keywords: External rays,
local connectedness,
robustness and Julia sets.