Convex hulls of polynomial Julia sets

Malgorzata Stawiska

paper

Convex hulls of polynomial Julia sets

arXiv:2004.12521

Abstract

We prove P. Alexandersson's conjecture that for every complex polynomial $p$ of degree $d \geq 2$ the convex hull $H_{p}$ of the Julia set $J_{p}$ of $p$ satisfies $p^{- 1} (H_{p}) \subset H_{p}$ . We further prove that the equality $p^{- 1} (H_{p}) = H_{p}$ is achieved only if $p$ is affinely conjugated to the Chebyshev polynomial $T_{d}$ of degree $d$ , to $- T_{d}$ or a monomial $c z^{d}$ with $∣ c ∣ = 1$ .