Dynamical complexity of Anosov systems driven by a quasi-periodic force

Consider C² Anosov systems on a compact manifold driven by a quasi-periodic force. We study their dynamical complexity on various levels from the perspectives of both path-wise dynamics and stochastic processes. Assuming that these systems are non-wandering (i.e., every point in the phase space is non-wandering), we prove a set of results: (1) the existence of abundance of random periodic points; (2) a random Livžic theorem; (3) a random Mañé-Bousch-Conze-Guivarc’h lemma; (4) the existence of strong random horseshoes. Additionally, a concrete example constructed on a 2-dimensional torus is also given to uncover some interesting phenomena of the systems.