We develop a Kolmogorov-Arnold-Moser (KAM)-theoretic framework to analyze fractal transport barriers and chaotic phase-space structures in two-dimensional incompressible (areapreserving) flows with periodic and quasi-periodic time dependence. The invariant sets of the Poincar´e map-ranging from smooth KAM tori to their cantori (fractal remnants of broken invariant tori)-are used to characterize material transport barriers and the onset of chaotic mixing. We show how surviving invariant tori confine trajectories, while resonance growth and overlap generate hierarchically organized, self-similar island chains and fractal chaotic seas that promote transport across phase space. We further analyze strong KAM stability near twistless (degenerate) tori, deriving scaling laws for resonant widths that explain the enhanced robustness of these fractal barrier structures and the associated suppression of cross-barrier flux. The methodology is illustrated using an Arnold-Beltrami-Childress (ABC) flow and a planar Hamiltonian proxy, revealing the canonical transition from integrable dynamics to a mixed phase space characterized by fractal invariant sets embedded within chaotic regions as the perturbation strength increases. The results provide a principled route to controlling fractal transport pathways: by tuning wave amplitudes, wavenumbers, and phase speeds, one can either preserve fractal barriers for segregation or induce resonance overlap for rapid homogenization. Extensions to weak diffusion, stochastic perturbations, and three-dimensional kinematics are discussed as avenues for predicting the longevity and permeability of fractal transport barriers in realistic flow settings.
Mei et al. (2026) studied this question.