Gauss-Lucas Theorem in Polynomial Dynamics

Explore the Gauss-Lucas theorem in polynomial dynamics through this 39-minute lecture from the Simons Semester on Dynamics. Delve into adapted versions of the theorem to prove that for every complex polynomial p of degree d ≥ 2, the preimage of the convex hull of the Julia set under p is a subset of itself. Examine the positive resolution of Per Alexandersson's 2020 conjecture and investigate cases where this preimage equals the convex hull. Conclude by analyzing examples of convex hull behavior for Julia sets of non-polynomial rational maps.