Persistent patterns in periodically driven dynamics have been reported in a wide variety of contexts ranging from table-top and ocean-scale fluid mixing systems to the weak quantum-classical transition in open Hamiltonian systems. We illustrate a common framework for the emergence of these patterns by considering a simple measure of structure maintenance provided by the average radius of the scalar distribution in transform space. Within this framework, scaling laws related to both the formation and persistence of patterns in phase space are presented. Further, preliminary results linking the scaling exponents associated with the persistent patterns to the multifractal nature of the advective phase-space geometry are shown.