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 nonwandering）,we prove a set of results:（1） the existence of abundance of random periodic points;（2） a random Liv?ic theorem;（3） a random Ma?é-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.