On Dynamics of Asymptotically Minimal Polynomials

TL;DR

This paper investigates the dynamical behavior of asymptotically extremal polynomials related to planar sets, showing convergence of measures and Julia sets under certain conditions.

Contribution

It establishes new results on the convergence of Brolin measures and Julia sets for asymptotically minimal polynomials associated with planar sets.

Findings

Brolin measures converge to the equilibrium measure when zeros are bounded.

Filled Julia sets converge to the polynomial convex hull of E under regularity and proximity conditions.

Provides conditions under which dynamical properties of these polynomials are characterized.

Abstract

We study dynamical properties of asymptotically extremal polynomials associated with a non-polar planar compact set E. In particular, we prove that if the zeros of such polynomials are uniformly bounded then their Brolin measures converge weakly to the equilibrium measure of E. In addition, if E is regular and the zeros of such polynomials are sufficiently close to E then we prove that the filled Julia sets converge to polynomial convex hull of E in the Klimek topology.