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.

We explore connections among the regional proximal relation, the asymptotic relation, and the distal relation for a topological dynamical system with the shadowing property and show that if a Devaney chaotic system has the shadowing property then it is distributionally chaotic.

ABSTRACT. Let (X, T ) be a topological dynamical system. A pair of points (x, y) ∈ X 2 is called Banach proximal if for any ε > 0, the set {n ∈ Z + : d(T n x, T n y) < ε} has Banach density one. We study the structure of the Banach proximal relation. An useful tool is the notion of the support of a topological dynamical system. We show that a dynamical system is strongly proximal if and only if every pair in X 2 is Banach proximal. A subset S of X is Banach scrambled if every two distinct points in S form a Banach proximal pair but not asymptotic. We construct a dynamical system with the whole space being a Banach scrambled set. Even though the Banach proximal relation of the full shift is of first category, it has a dense Mycielski invariant Banach scrambled set. We also show that for an interval map it is Li-Yorke chaotic if and only if it has a Cantor Banach scrambled set.

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