Semigroups of branched mapping classes: dynamics and geometry

Suppose $P \subset S^2$ is a finite subset of the sphere. The  mapping class group $\mbox{Mod}(S^2,P)$ is the countable group of orientation-preserving homeomorphisms $f:(S^2, P) \to (S^2, P)$ with $f(P)\subset P$ , up to isotopy through homeomorphisms fixing $P$ pointwise. If we allow $f$ to be instead a branched covering whose branch values are contained in $P$ with $f(P) \subset P$, we obtain a countable semigroup. Its study has rich connections to one- and several-variable complex dynamics, to dynamics and geometry on Teichmueller and moduli spaces, and to the theory of self-similar groups. I will introduce the subject and discuss some recent developments.