`Combining Fuchsian reflection groups and quadratic polynomial dynamics (after Lee, Lyubich, Makarov and Mukherjee)'

Abstract: Two fundamentally different kinds of holomorphic dynamical
systems exist on the Riemann sphere:
1) Fuchsian or more generally Kleinian groups
2) Iteration of polynomials or rational maps.
They have been studied for more than a century starting with Poincare,
Fatou and Julia amongst others.

A dictionary between the fundamental structures of these two kinds of
dynamical systems was brought into focus by Sullivan in the 80's and
subsequently by McMullen. However a common
framework for dealing with both is lacking. A couple of recent papers by
Lee,  Lyubich, Makarov and Mukherjee changes that and provides a natural
class of examples where Fuchsian reflection groups on the one hand are
combined with the dynamics of $z \to z^2 +c$ on the other. This will be an
expository talk where we shall first describe the dictionary, say what is
meant by combining dynamical systems and conclude with a description of
the results of the authors.