Explore the fascinating intersection of complex dynamics and surface homeomorphisms in this 29-minute lecture from the Simons Semester on Dynamics. Delve into a novel connection between quadratic polynomial dynamics on the complex plane and surface homeomorphism dynamics. Examine the behavior of quadratic polynomials on Hubbard trees and investigate the possibility of extending this behavior to generalized pseudo-Anosov homeomorphisms. Learn about a newly developed criterion for determining which quadratic polynomials allow for such extensions, bridging the gap between holomorphic dynamics and surface topology.