Speaker: Lasse Rempe (University of Manchester)
TITLE: Maverick capacity
ABSTRACT. This talk concerns the dynamics of a transcendental entire function f; that is, a non-polynomial holomorphic self-map of the complex plane. The Fatou set F(f) consists of those points near which the dynamics is stable; that is, points sufficiently close to a starting value in F(f) will remain close to the orbit of this value under iteration.
We are interested in wandering domains; that is, connected components of the Fatou set that are not eventually periodic. By a famous theorem of Sullivan, rational functions do not have wandering domains, but these domains may exist for transcendental entire functions. Suppose that U is a wandering domain on f which is escaping; that is, the iterates tend to infinity on U. Rippon and Stallard proved that escaping points have full harmonic measure on the boundary of U. In 2014, Bishop asked whether non-escaping points on the boundary (which we call \emph{maverick points}) even have logarithmic capacity zero with respect to U, which is a much stronger condition.
We give an example that shows that the answer is negative in general, but prove that the answer is positive when all iterates are univalent on U. This is joint work with Martí-Pete and Waterman.