Some Dynamical Properties Of Dendrite

In recent years, dynamical systems on dendrites, such as the equicon-tinuity, the separation index, the convergence of the orbits, the minimal sets and the depth of the centre of a dendrite map, have been studied by many authors. Thus this article mainly aims at the study of the two aspects of den-drites.A continuum is a non-empty compact connected metric space. A den-drite is a locally connected continuum which contains no simple closed curve. This article mainly has the following result.In chapter 3, Let D be a dendrite with finite branch points and f:Dâ†'D be continuous. Denote by R(f) and Ω(f) the set of recurrent points and the set of non-wandering points of f respectively. Let Ωo(f)= D and Ωn(f)= Ω(f|Ωn-1(f)) for all n ∈ N. The minimal m ∈ N∪{∞} such that (f) Ωm+1(f) is called the depth of f. In this note, we show thatΩ3(f)= R(f) and the depth of f is at most 3. Furthermore, we show that there exist a dendrite D with finite branch points and f ∈Co(D) such thatΩ3(f)= R(f) Ω2(f).In chapter 4, In addition, Let (T,d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by ω(x,f) the ω-limit set of x under f. Write Ω(x,f)={y|there exist a sequence of points xk ∈T and a sequence of positive integers n1<n2<… such that and For the equicontinuous study on dendrite, we show that the following statements are equivalence:(1)f is equicontinuous. (2) ω)(x,f)= Ω.(x,f) for any x∈T. (3) Ω(x,f) is a periodic orbit for any x ∈ T. (4) ∩∞n=1fn(T)= p(f), Card(ω(x,f))<∞ for any x ∈T and map h: x â†'ω(x,f) (x ∈T) is continuous, we also construct and prove the other two special dendrite:(1) there exist a dendrite D and f ∈ Co(D), such that P(f)=Ω(f)≠D= CR(f) and fn has no turbulent, for any n∈EN. (2) there exist a dendrite D and f ∈Co(D), with x,y ∈ D, such that x(?) Sα(x,f), when x ∈ Sα (y, f).