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In many dynamical systems, countably infinitely many invariant tori co-exist.The occurrence of quasiperiodicity on any one of these tori is sometimessufficient to establish strong global properties, like dense trajectories andperiodic points. In this paper, we establish sufficient conditions for acountably infinite collection of parameterized circle diffeomorphisms to havequasiperiodic behavior on at least one of the circles, for a full Lebesguemeasure set of the parameter values. As an application, we study parameterizedfamilies of skew-product maps on the torus and prove sufficient conditions forthe existence of at least on quasiperiodic circle for Lebesgue-almost everyparameter value.

Almost-sure quasiperiodicity in countably many co-existing circles