Some Dynamical Properties Of Continuous Maps On Sin(1/x) Continuum

The research of the dynamical properties of continuous maps on con-tinuum have been played an important part in dynamical systems.In the continuum theory,both sin(1/x)continuum and Warsaw circle are classic examples.In recent years,many scholars have studied the dynamical prop-erties of continuous maps on the Warsaw cycle,while there are few papers about the dynamical properties of continuous maps on sin(1/x)continuum.In this thesis,it is mainly researched the one-side interval property、PR prop-erty、pointwise chain recurrence and equicontinuity of continuous maps on sin(1/x)continuum.Let S be the sin(1/x)continuum and f:S→S be a continuous map,where S=L1∪L2,L1={(x,y)∈R2 |x=0,-1≤y≤1},L2 = {(x,sin(1/x))∈R2|0<x≤1}.If f(L1)(?)L1,then write f1 = f|L1.If f(L2)(?)L2,then write f2=f|L2.Let Fix(f),P(f),Pn(f)and R(f)denote the set of all of fixed points,periodic points,n periodic points,and recurrent points of f,respec-tively.There are mainly the following conclusions in the thesis:1、Interval without periodic points is a one-side interval,and P(f)=R(f).2、If f is pointwise chain recurrent,then(1)if Fix(f)is connected,then f is identical;and(2)if Fix(f)is disconnected,then f is turbulent either Fix(f1)or Fix(f2)is nondegenerate disconnected;f is not turbulent whenever Fix(f1)= L1,Fix(f2)=a,a∈L2 and(L2-{a})∩P(f2)=0.3、If f is an equicontinuous map,then P(f)= Fix(f2),both Fix(f)and Fix(f2)are connected.Furthermore,if Fix(f)is non-degenerate,then Fix(f)=P(f).4、Let f(L1)(?)L1,f(L2)(?)L2.If f is equicontinuous,then Fix(f2)=∩ n≥1 fn(S),and Fix(f2)is connected.Moreover,if P(f2)≠0,then there existsa ∈(0,1]such that P(f)= Fix(f)= {(x,y):(x,y)∈S and x ≤a}.5、(1)If f(S)(?)Li for each i ∈{1,2},and Fix(f)=∩n≥1n=(S),then fis equicontinuous;and(2)if f(Li)(?)Li for each i=1,2,Fix(f2)=∩n≥1 fn(S)and Fix(f|L2)≠0,then f is equicontinuous.