This paper reports the experimental observation of anomalous transport in a kicked quasicrystal, a system where particles exhibit non-trivial transport behavior. The Hamiltonian of the system is designed to have a rich phase diagram, including fully localized and delocalized phases, as well as a nested pattern of localized, delocalized, and multifractal states, which gives rise to anomalous transport. The experiment uses cold atoms and new Floquet engineering techniques to expand the accessible phase diagram by five orders of magnitude. The researchers map the transport properties across the phase diagram, observing disorder-driven re-entrant delocalization and sub-ballistic transport. They provide a theoretical explanation for these phenomena based on eigenstate multifractality. The study demonstrates the first experimental exploration of the complex phase diagram of a prototypical model of localization in driven quantum matter, highlighting the potential of apodized Floquet engineering techniques for studying other long-range Hamiltonians and phenomena.This paper reports the experimental observation of anomalous transport in a kicked quasicrystal, a system where particles exhibit non-trivial transport behavior. The Hamiltonian of the system is designed to have a rich phase diagram, including fully localized and delocalized phases, as well as a nested pattern of localized, delocalized, and multifractal states, which gives rise to anomalous transport. The experiment uses cold atoms and new Floquet engineering techniques to expand the accessible phase diagram by five orders of magnitude. The researchers map the transport properties across the phase diagram, observing disorder-driven re-entrant delocalization and sub-ballistic transport. They provide a theoretical explanation for these phenomena based on eigenstate multifractality. The study demonstrates the first experimental exploration of the complex phase diagram of a prototypical model of localization in driven quantum matter, highlighting the potential of apodized Floquet engineering techniques for studying other long-range Hamiltonians and phenomena.