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Topological size of scrambled sets

François BlanchardWen HuangL'ubomír Snoha — 2008

Colloquium Mathematicae

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 l i m i n f n d ( f ( x ) , f ( y ) ) = 0 and l i m s u p n d ( f ( x ) , f ( y ) ) > 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...

Sequence entropy pairs and complexity pairs for a measure

Wen HuangAlejandro MaassXiangdong Ye — 2004

Annales de l’institut Fourier

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...

Topological disjointness from entropy zero systems

Wen HuangKyewon Koh ParkXiangdong Ye — 2007

Bulletin de la Société Mathématique de France

The properties of topological dynamical systems ( X , T ) which are disjoint from all minimal systems of zero entropy, 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 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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