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.

THE APOTP AND NON-WANDERING HOMEOMORPHISM

We introduce the concept of asymptotic pseudo orbit tracing property (APOTP) and obtain a new condition by the APOTP for which a homeomor-phism is a non-wandering homeomorphism.

Non-wandering points and the depth for graph maps

Let G be a graph and f:G→G be continuous.Denote by R(f) andΩ(f) the set of recurrent points and the set of non-wandering points of f respectively.LetΩ_0(f) = G andΩ_n(f)=Ω(f|_(Ω_(n-1)(f))) for all n∈N.The minimal m∈NU {∞} such thatΩ_m(f)=Ω_(m+1)(f) is called the depth of f.In this paper,we show thatΩ_2 (f)=(?) and the depth of f is at most 2.Furthermore,we obtain some properties of non-wandering points of f.

Non-wandering sets of the powers of maps of a tree

Let T be a tree and let Ω ( f ) be the set of non-wandering points of a continuous map f: T→ T. We prove that for a continuous map f: T→ T of a tree T: ( i) if x∈ Ω( f) has an infinite orbit, then x∈ Ω( fn) for each n∈ ?; (ii) if the topological entropy of f is zero, then Ω( f) = Ω( fn) for each n∈ ?. Furthermore, for each k∈ ? we characterize those natural numbers n with the property that Ω(fk) = Ω(fkn) for each continuous map f of T.

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.

Non-Wandering Sets of the Powers of Dendrite Maps

Let （X, d） be a metric space and f be a continuous map from X to X. Denote by EP（f） and Ω（f） the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold： （1） If x ∈ Ω（f） - Ω（f^n） for some n ≥ 2, then x ∈ EP（f）. （2） Ω（f） is contained in the closure of EP（f）. The aim of this note is to show that the above results do not hold for maps of dendrites D with Card（End（D）） = No （the cardinal number of the set of positive integers）.

Non-wandering Expanding Maps on Branched 1-Manifolds and Smale–Williams Solenoids

In this paper, we give a criterion of whether there are non-wandering expanding maps on a given branched 1-manifold. As an application, we give an example of a dynamic on a 3-manifold having non-zero first Betti number, the non-wander sets of which are two Smale–Williams solenoids.

A SIMPLER PROOF OF THE EXTENDED C^1 CLOSING LEMMA

The C^1 Closing Lemma and its extension have been discussed in detail in Refs. [1--4]. It is Professor Liao Shantao who first gave a correct proof of this lemma and its extension. In this note, using the theorem on the existence of the Nfold-undisturbing, progressive and minute movement under a sequence of linear transformations in [5], and making a meticulous analysis of C^1 norms of C^1 vector fields on manifolds, we give a simpler proof of the extended C^1 Closing Lemma.