In 2003, Bressan proposed a conjecture on the mixing efficiency of incompressible flows, which remains open. This talk surveys progress toward resolving Bressan’s mixing conjecture and presents a new result confirming its asymptotic validity for time-periodic velocity fields. We accomplish this by adapting dynamical systems tools to the non-smooth framework of DiPerna-Lions flows. Furthermore, we discuss links to bounds on metric entropy and extensions of the Ruelle inequality.