A potential-theoretic characterization of polynomials in holomorphic dynamics in one variable}

Abstract: In 1960s Hans Brolin initiated systematic application of potential-theoretic methods in the dynamics of holomorphic maps. Among other things, he proved the now-famous equidistribution theorem: for a complex 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 (with pole at infinity) for the Julia set $J_f$ of $f$. To a general rational map $f$ of degree $d \geq 2$ on the Riemann sphere $\mathbb{CP}^1$ one can associate another measure $\mu$, called the balanced measure. It is supported on the Julia set for $f$ and satisfies $f*\mu=d \cdot \mu$. Since it also can be obtained as the limit of preimages of quite general probabilistic measures on $\mathbb{CP}^1$ (thanks to the results of M. Lyubich and independently Freire-Lopes-R. Ma\~ne from 1980s), a question arises whether it always equals the equilibrium measure for $J_f$ (when the latter notion makes sense). Several mathematicians noticed that equality of these two measures (under suitable assumptions on $f$) implies that $f$ is a polynomial. However, all ``proofs'' of this implication from before 1990s contained gaps. The proof by S. Lalley from 1992 was fully successful, but it was based on the theory of Brownian motions. In this talk, I will present a general version of this implication with a proof using mainly classical and weighted potential theory: Let $f: \mathbb{CP}^1 \to \mathbb{CP}^1$ be a rational function of degree $d \geq 2$ whose Julia set does not contain the point $\infty$. The following are equivalent: (i) $f \circ f$ is a polynomial; (ii) the balanced measure for $f$ and the equilibrium measure for the Julia set $J_f$ with pole at infinity are equal. This is joint work with Y\^usuke Okuyama from Kyoto Institute of Technology.

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

Series comments: Please e-mail R.Halburd@ucl.ac.uk for the Zoom link. Also, please let me know whether you would like to be added to the mailing list to automatically receive links for future talks in CAvid.