abstract: While dynamical systems with positive topological entropy have been widely studied, the zero entropy cases are much less well understood. This talk will be dedicated to a weaker measure of complexity, the polynomial entropy, for which we will first recall the definitions and main properties, and then discuss two recent applications: the classification of conjugacy classes of Brower homeomorphisms of the plane (following Hauseux and Le Roux) and a dynamical version of the Birkhoff conjecture of convex billiards.