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In a class of periodically driven systems, multifractal states innon-equilibrium conditions and robustness of dynamical localization when thedriving is made aperiodic have received considerable attention. In this paper,we explore a family of one-dimensional Aubry-Andr\'e-Harper models that arequasi-periodically kicked with protocols following different binaryquasi-periodic sequences, which can be realized in ultracold atom systems. Therelationship between the systems' localization properties and the sequences'mathematical features is established utilizing the Floquet theorem and theBaker-Campbell-Hausdorff formula. We investigate the multifractality andpre-thermalization of the eigenstates of the unitary evolution operatorcombined with an analysis of the transport properties of initially localizedwave packets. We further contend that the quasi-periodically kickedAubry-Andr\'e-Harper model provides a rich phase diagram as the periodic casebut also brings the range of parameters to observe multifractal states andpre-thermalization to a regime more amenable to experiments.

Multifractality and pre-thermalization in the quasi-periodically kicked Aubry-Andr\'e-Harper model