Algebraic Dynamics Seminar: The question mark function, welding and complex dynamics

We show that infinitely many cusped Julia sets arising in complex dynamics are essentially the same as the Jordan curve W(?) defined using Minkowski’s question mark function as a conformal welding homeomorphism. The proof of this universality rests on a general result in real dynamics: the function ?(x) arises from any topological conjugacy between parabolic and hyperbolic dynamical systems on S^1 (subject to the condition of geometric finiteness). We also show that Aut(?) is very rich; for example, it contains a copy of PGL(2,Q). As a corollary we show that the set of welding maps on S^1 is not closed under composition.

for more information