Let X be a compact metric space, and let f: X→X be transitive with X infinite. We show that each asymptotic class (or the stable set Ws (x) for each x ∈ X) is of first category and so is th asymptotic relation. Moreover, we prove that if the proximal relation is dense in a neighbourhood of some point in the diagonal then f is chaotic in the sense of Li-Yorke. As applications we obtain that if f contains a periodic point, or f is 2-scattering, then f is chaotic in the sense of Li-Yorke. Thus, chaos in the sense of Devaney is stronger than that of Li-Yorke. Elsevier Science B.V. All rights reserved.

A homeomorphism on a metric space (X; d) is completely scrambled if for each x ≠ y ∈ X, lim supn　sup→+∞d (fn (x), fn (y)) > 0 and lim infn→+∞C1 d (fn (x), fn (y)) = 0. We study the basic properties of completely scrambled homeomorphisms on compacta and show that there are 'many' compacta admitting completely scrambled homeomorphisms, which include some countable compacta (we give a characterization), the Cantor set and continua of arbitrary dimension.

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.

measure-preserving transformation (respectively a topological system) is null if the metric (respectively topological) sequence entropy is zero for any sequence. Kushnirenko has shown that an ergodic measure-preserving transformation has a discrete spectrum if and only if it is null. We prove that for a minimal system this statement remains true modulo an almost one-to-one extension. It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, some necessary conditions for a transitive non-minimal system to be null are obtained.

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 thatthere is a distal system in M.

A dynamical system (X, T) is F-transitive if for each pair of open and non-empty subsets U and V of X, N (U,V) = {n ∈ Z+: U ∩ T−nV ≠ Ø} ∈ F-where F is a collection of subsets of Z+ that is hereditary upward. (X, T) is F-mixing if (X×X, T×T) is F-transitive. For a subset S of Z+, (x, y ∈ X×X is S-proximal if lim inf Sn →+∞ d (Tn (x), Tn (y)) = 0 and the S-proximal cell PS (x) is the set of points that are S-proximal to x ∈ X. We show that if (X, T) is F-mixing, then for each S ∈ kF (the dual family of F) and x ∈ X, PS (x) is a dense Gδ subset of X, and when (X, T) is minimal and Fis a filter the reciprocal is true. Moreover, other conditions under which the reciprocal is true are obtained. Finally the structure of proximal cells for F-mixing systems is discussed, and a newand simpler proof of the Xiong-Yang theorem is presented.

By a dynamical system we mean a pair (X, T), where X is a compact metric space and T: X→X is surjective and continuous. We study weak disjointness in topological dynamics. (X, T) is scattering iff it is weakly disjoint from all minimal systems and (X, T) is strongly scattering iff it is weakly disjoint from all E-systems, i.e. transitive systems having invariant measures with full support. It is clear that a weakly mixing system is strongly scattering and the latter is scattering. An existential proof of scattering and a non-weakly mixing example is obtained by Akin and Glasner (2001 J. Anal. Math. 84 243-86). In this paper, we will give an explicit example which is strongly scattering and not weakly mixing. We also define extreme scattering, weak scattering and study the relationships of the various definitions.By a dynamical system we mean a pair (X, T), where X is a compact metric space and T: X → X is surjective and continuous. We study weak disjointness in topological dynamics. (X, T) is scattering iff it is weakly disjoint from all minimal systems and (X, T) is strongly scattering iff it is weakly disjoint from all E-systems, i.e. transitive systems having invariant measures with full support. It is clear that a weakly mixing system is strongly scattering and the latter is scattering. An existential proof of scattering and a non-weakly mixing example is obtained by Akin and Glasner (2001 J. Anal. Math. 84 243-86). In this paper, we will give an explicit example which is strongly scattering and not weakly mixing. We also define extreme scattering, weak scattering and study the relationships of the various definitions.

In [BGH] the author show that for a given topological system (X; T) and an open cover U there is an invariant measuresuch that inf h (T; P), htop (T; U), where infimum is taken over all partitions which are finer than U. We prove in this paper that if is an invariant measure and inf h

(X, T) be a topological dynamical system and let µ be a T-invariant probability measure on X. In this paper, we study two properties of the notions of measure theoretical entropy for a measurable cover U, h+µ (U, T) and h−µ (U, T) introduced by P. P. Romagnoli (Ergod. Th. & Dynam. Sys. 23 (2003), 1601-1610). The main result of the paper states that entropy pairs for the measure µ can be defined using either h+µ or h−µ. We also prove that both h+µ and h−µ have an ergodic decomposition and we use it to provea local Abramov formula for h−µ.