A subset S of a topological dynamical system (X,f) containing at least two points is called a scrambled set if for any x,y ∈ S with x ≠ y one has
$limin{f}_{n\to \infty}d(f\u207f\left(x\right),f\u207f\left(y\right))=0$ and $limsu{p}_{n\to \infty}d(f\u207f\left(x\right),f\u207f\left(y\right))>0$,
d being the metric on X. The system (X,f) is called Li-Yorke chaotic if it has an uncountable scrambled set.
These notions were developed in the context of interval maps, in which the existence of a two-point scrambled set implies Li-Yorke chaos and many other chaotic properties. In the present paper we address several questions about scrambled...

In this paper we explore topological factors in between the Kronecker factor and the
maximal equicontinuous factor of a system. For this purpose we introduce the concept of
sequence entropy $n$-tuple for a measure and we show that the set of sequence entropy
tuples for a measure is contained in the set of topological sequence entropy tuples [H-
Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we
introduce a weak notion and a strong notion of complexity pair for a...

The properties of topological dynamical systems $(X,T)$ which are disjoint from all minimal systems of zero entropy, ${\mathcal{M}}_{0}$, are investigated. Unlike the measurable case, it is known that topological $K$-systems make up a proper subset of the systems which are disjoint from ${\mathcal{M}}_{0}$. We show that $(X,T)$ has an invariant measure with full support, and if in addition $(X,T)$ is transitive, then $(X,T)$ is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists a thickly syndetic...

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