CHAIN RECURRENT POINTS AND TOPOLOGICAL ENTROPY OF A TREE MAP

Let f be a tree map,P(f) the set of periodic points of f and CR(f) the set of chain recurrent points of f. In this paper,the notion of division for invariant closed subsets of a tree map is introduced.It is proved that: (1) f has zero topological entropy if and only if for any x∈CR(f)-P(f) and each natural number s the orbit of x under f s has a division; (2) If f has zero topological entropy,then for any x∈CR(f)-P(f) the ω-limit set of x is an infinite minimal set.

Banach Upper Density Recurrent Points of C^0-flows

Let X denote a compact metric space with distance d and F ： X×R→ X or Ft ： X→X denote a C0-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C0-flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points.

Distributional Chaos Occurring on the Set of Proper Positive Upper Banach Density Recurrent Points of One-sided Symbolic Systems

The purpose of this paper is to show that for one-sided symbolic systems,there exists an uncountable distributionally scrambled set contained in the set of proper positive upper Banach density recurrent points.

NON-WANDERING SET OF A CONTINUOUS GRAPH MAP

The non-wandering set Ω(f) for a graph map f is investigated. It is showed that Ω(f) is contained in the closure of the set ER(f) of eventually recurrent points of f and ω-limit set ω(Ω(f)) of Ω(f) is contained in the closure of the set R(f) of recurrent points of f.

Relative Broken Family Sensitivity

Letπ:(X,T)→(Y,S)be a factor map between two topological dynamical systems,and F_(a) Furstenberg family of Z.We introduce the notion of relative broken F-sensitivity.Let Fs(resp.Fpubd,Finf)be the families consisting of all syndetic subsets(resp.positive upper Banach density subsets,infinite subsets).We show that for a factor mapπ:(X,T)→(Y,S)between transitive systems,πis relatively broken F-sensitive for F=Fs or Fpubd if and only if there exists a relative sensitive pair which is an F-recurrent point of(R_(π),T^((2)));is relatively broken Finf-sensitive if and only if there exists a relative sensitive pair which is not asymptotic.For a factor mapπ:(X,T)→(Y,S)between minimal systems,we get the structure of relative broken F-sensitivity by the factor map to its maximal equicontinuous factor.

Recurrence of Transitive Points in Dynamical Systems with the Specification Property

Let T ： X →X be a continuous map of a compact metric space X. A point x E X is called Banach recurrent point if for all neighborhood V of x, （n ∈ N ： T^n（x） ∈ V} has positive upper Banach density. Denote by Tr（T）, W（T）, QW（T） and BR（T） the sets of transitive points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points of （X, T）. If （X, T） has the specification property, then we show that every transitive point is Banach recurrent and O≠ W（T） n Tr（T） ≠ W＊（T） ∩ Tr（T） ≠ QW（T） ∩ Tr（T） ≠ BR（T） ∩ Tr（T）, in which W＊（T） is a recurrent points set related to an open question posed by Zhou and Feng. Specifically the set Tr（T） M W＊（T） ／ W（T） is residual in X. Moreover, we construct a point x E BR ／ QW in symbol dynamical system, and demonstrate that the sets W（T）, QW（T） and BR（T） of a dynamical system are all Borel sets.

Special α-limit points and unilateral γ-limit points for graph maps

Let G be a graph and f ： G → G be a continuous map. Denote by P（f）, R（f）, SA（f） and UF（f） the sets of periodic points, recurrent points, special α-limit points and unilateral γ-limit points of f, respectively. In this paper, we show that R（f） SA（f） = UP（f） ∪ P（f） R（f）.

Multi-recurrence and van der Waerden systems

We explore recurrence properties arising from dynamical approach to the van der Waerden theorem and similar combinatorial problems. We describe relations between these properties and study their consequences for dynamics. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity of topologically weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.

Topological Structure of Non-wandering Set of a Graph Map

Let G be a graph （i.e., a finite one-dimensional polyhedron） and f ： G → G be a continuous map. In this paper, we show that every isolated recurrent point of f is an isolated non-wandering point; every accumulation point of the set of non-wandering points of f with infinite orbit is a two-order accumulation point of the set of recurrent points of f; the derived set of an ω-limit set of f is equal to the derived set of an the set of recurrent points of f; and the two-order derived set of non-wandering set of f is equal to the two-order derived set of the set of recurrent points of f.

Topological Entropy of a Graph Map

Let G be a graph and f： G → G be a continuous map. Denote by h（f）, P（f）, AP（f）, R（f） and w（x, f） the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the w-limit set of x under f, respectively. In this paper, we show that the following statements are equivalent： （1） h（f） 〉 O. （2） There exists an x ∈ G such that w（x, f） ∩ P（f） ≠θ and w（x, f） is an infinite set. （3） There exists an x ∈ G such that w（x, f） contains two minimal sets. （4） There exist x, y ∈G such that w（x, f） - w（y, f） is an uncountable set andw（y,f）∩w（x,f）≠θ. （5） There exist anx C Gand a closed subset A w（x,f） with f（A） A such that w（x,f） - A is an uncountable set. （6） R（f） - nP（f） ≠θ. （7） f｜P（f） is not pointwise equicontinuous.

CHAOS AND ORDER OF INVERSE LIMIT SPACE FOR A GRAPH MAP

With the method of inverse limit, the author obtains several criteria of chaos of piecewise monotone continuous maps on finite graphs.

THE LIMIT SET OF A CONTINUOUS SELF-MAP OF THE TREE

Let f denote a continuous map of a tree T to itself. A point x ∈ T is called a 7-limit point of f if it is both an ω-limit point and an α-limit point. In the present paper, we show that (1) Ω-Γ is countable, (2) A -Γ and P - Γ are either empty or countably infinite, where P denotes the closure of the set of periodic points P.