Department of Mathematics

A characterization of polynomials in holomorphic dynamics in one complex variable using potential theory

  • Date: 3/23/2017 at 1:15 PM
  • Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
  • Lecturer: Margaret Stawiska-Friedland (University of Michigan-Ann Arbor)
  • Contact person: Konstantinos Dareiotis
  • Seminarium

Analysis and probability seminar

ABSTRACT: In 1960's Hans Brolin initiated systematic application of potential-theoretic methods in the dynamics of holomorphic polynomials. Among other things, he proved the now-famous equidistribution theorem: for a polynomial $f$ of degree greater than $1$ the preimages, under successive iterates of $f$, of a Dirac measure at an arbitrary point of the complex plane (except at most two so-called exceptional points) converge weakly to the equilibrium measure of the Julia set for $f$. In 1980's a similar result (about convergence of preimages of quite general probabilistic measures) was proved for a rational map $f$ of degree greater than $1$. The limit measure obtained in this case (called the balanced measure) is also supported on the Julia set for $f$, but does not have to be its equilibrium measure. In fact, A.O. Lopes proved (using dynamical properties of Julia sets) that equality of these two measures (under suitable assumptions on $f$, also making precise the notion of the equilibrium measure for the Julia set) implies that $f$ is a polynomial. In this talk I present a proof of Lopes's theorem (under slightly weaker assumptions) using classical and weighted potential theory. It is joint work with Yusuke Okuyama from Kyoto Institute of Technology.