Math Sem: Lukas Geyer (MSU), Stability of indifferent fixed points and the Hadamard 3-circle theorem

Abstract:

Questions about the stability of a fixed point $z_0$ of an analytic map $f$ in complex dynamics are closely connected to properties of the multiplier $\lambda = f'(z_0)$. Particularly interesting and difficult is the case of an irrationally indifferent fixed point, where $\lambda = e^{2\pi i \alpha}$ is on the unit circle with irrational rotation number $\alpha$. In this case, answers depend very much on number-theoretic properties of the rotation number, and several questions remain open.

In this talk I will present applications of a beautiful classical result from complex analysis, the Hadamard three-circle theorem, to this question, a technique which was first used by Hubert Cremer in 1948 in the context of dynamics of entire functions.

This talk will be accessible to graduate students with a background in basic complex analysis, and one of the aims is to advertise the power of some classical complex analysis results which are typically not taught in the standard graduate classes.