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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.

In Blanchard et al (Topological complexity. Ergod. Th. & Dynam. Sys.20 (2000), 641–662), the authors introduced the notion of scattering and a weaker notion of 2-scattering. It is an open question whether the two notions are equivalent. The question is answered affirmatively in this paper. Using the complexity function of an open cover along some sequences of natural numbers, we characterize mild mixing, strong scattering and scattering. We show that mildly mixing (respectively strongly mixing) systems are disjoint from minimal uniformly rigid (respectively minimal rigid) systems.Moreover, assuming minimality we show that a dynamical system is full-scattering (respectively mildly mixing or weakly mixing) if and only if it is strongly mixing (respectively IP*-transitive or $\mathcal{D}$-transitive), where $\mathcal{D}$ is the collection of subsets of $\mathbb{Z}_+$ with the lower Banach density 1.

For an invariant measure µ in a topological dynamics, notions of µ-sensitivity, µ-complexity and µ-equicontinuity are introduced and investigated.

Abstract. Furstenberg showed that if two topological systems (X, T ) and (Y, S) are disjoint, then one of them, say (Y, S), is minimal. When (Y, S) is nontrivial, we prove that (X, T ) must have dense recurrent points, and there are countably many maximal transitive subsystems of (X, T ) such that their union is dense and each of them is disjoint from (Y, S). Showing that a weakly mixing system with dense periodic points is in M ⊥ , the collection of all systems disjoint from any minimal system, Furstenberg asked the question to characterize the systems in M ⊥ . We show that a weakly mixing system with dense regular minimal points is in M ⊥ , and each system in M ⊥ has dense minimal points and it is weakly mixing if it is transitive. Transitive systems in M ⊥ and having no periodic points are constructed. Moreover, we show that there is a distal system in M ⊥ .Recently, Weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic. We construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing. §1. Introduction By a topological dynamical system (TDS for short) (X, T ) we mean a compact metric space X with a continuous surjective map T from X to itself. Recall that (X, T ) is transitive if for each pair of open (i.e., nonempty and open) subsets U andIt is well known if (X, T ) is transitive, then the set of transitive points is a dense G δ set (denoted by T ran T ) and if T ran T = X, we say that (X, T ) is minimal. For a minimal system (X, T ) each point of X is called a minimal point.

The local properties of entropy for a countable discrete amenable group action are studied. For such an action, a local variational principle for a given finite open cover is established, from which the variational relation between the topological and measure-theoretic entropy tuples is deduced. While doing this it is shown that two kinds of measure-theoretic entropy for finite Borel covers coincide. Moreover, two special classes of such an action: systems with uniformly positive entropy and completely positive entropy are investigated.