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.