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ABSTRACT. Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal.Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the later property must have zero entropy.

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In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's Möbius disjointness conjecture holds for any system for which every invariant Borel probability measure has sub-polynomial measure complexity.Moreover, it is proved that the following classes of topological dynamical systems (X, T ) meet this condition and hence satisfy Sarnak's conjecture: (1) Each invariant Borel probability measure of T has discrete spectrum.(2) T is a homotopically trivial C ∞ skew product system on T 2 over an irrational rotation of the circle. Combining this with the previous results it implies that the Möbius disjointness conjecture holds for any C ∞ skew product system on T 2 . (3) T is a continuous skew product map of the form (ag, y + h(g)) on G × T 1 over a minimal rotation of the compact metric abelian group G and T preserves a measurable section. (4) T is a tame system.