Mean equicontinuity and mean sensitivity

 In this talk, we study equicontinuity and sensitivity in the mean sense.
We show 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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