Quasi-conformal Surgery

The technique of quasi-
conformal surgery in
holomorphic dynamics was
initiated by Sullivan,
Douady, Hubbard and
Shishikura in the early
eighties. The method is to
create new dynamical systems
out of some given ones, by
changing not only the
dynamical plane (through
cutting and sewing) and the
map defining the dynamical
system, but also the complex
structure of the new
dynamical plane. The theory
of quasi-conformal mappings
is the basic tool. Surgery
techniques appear to be
particularly successful, when
two families of maps can be
related in such a way that
dynamical similarities are
transferred to similarities
between structures in the
corresponding parameter
spaces. Homeomorphisms
between p/q-limbs of the
Mandelbrot set, with fixed
denominator q, have been
obtained. Generalizations are
currently being investigated.