Let I=[0,1],c_1,c_2 ∈(0,1)with c_1<c_2 and f:I→I be a continuous map satisfying:f|_[0,c_1] and f|_[c_2,1] are both strictly increasing and f|_[c_1,c_2]is strictly decreasing.Let A ={z ∈[0,c_1]|f(x)=x}, a=maxA,a_...Let I=[0,1],c_1,c_2 ∈(0,1)with c_1<c_2 and f:I→I be a continuous map satisfying:f|_[0,c_1] and f|_[c_2,1] are both strictly increasing and f|_[c_1,c_2]is strictly decreasing.Let A ={z ∈[0,c_1]|f(x)=x}, a=maxA,a_1=max(A\{a}),and B={x∈[c_2,1]|f(x)=x},b=minB,b_1=min(B\{b}).Then the in- verse limit(I,f)is an arc if and only if one of the following three conditions holds: (1)If c_1<f(c_1)≤c_2(resp.c_1≤f(c_2)<c_2),then f has a single fixed point,a period two orbit, but no points of period greater than two or f has more than one fixed point but no points of other periods,furthermore,if A≠φ and B≠φ,then f(c2)>a(resp.f(c_1)<b). (2)If f(c_1)≤c_1(resp.f(c_2)≥c_2),then f has more than one fixed point,furthermore,if B≠φ and A\{a}≠φ,f(c_2)≥a or if a_1<f(c_2)<a,f^2(c_2)>f(c_2),(resp.f has more than one fixed point,furthermore,if A≠φ and B\{b}≠φ,f(c_1)≤b or if b<f(c_2)<b_1,f^2(c_1)<f(c_1)). (3)If f(c_1)>c_2 and f(c_2)<c_1,then f has a single fixed point,a single period two orbit lying in I\(u,v)but no points of period greater than two,where u,v ∈[c_1,c_2] such that f(u)=c_2 and f(v)=c_1.展开更多
基金Supported by the National Natural Science Foundation of China(No.19961001,No.60334020)Outstanding Young Scientist Research Fund.(No.60125310)
文摘Let I=[0,1],c_1,c_2 ∈(0,1)with c_1<c_2 and f:I→I be a continuous map satisfying:f|_[0,c_1] and f|_[c_2,1] are both strictly increasing and f|_[c_1,c_2]is strictly decreasing.Let A ={z ∈[0,c_1]|f(x)=x}, a=maxA,a_1=max(A\{a}),and B={x∈[c_2,1]|f(x)=x},b=minB,b_1=min(B\{b}).Then the in- verse limit(I,f)is an arc if and only if one of the following three conditions holds: (1)If c_1<f(c_1)≤c_2(resp.c_1≤f(c_2)<c_2),then f has a single fixed point,a period two orbit, but no points of period greater than two or f has more than one fixed point but no points of other periods,furthermore,if A≠φ and B≠φ,then f(c2)>a(resp.f(c_1)<b). (2)If f(c_1)≤c_1(resp.f(c_2)≥c_2),then f has more than one fixed point,furthermore,if B≠φ and A\{a}≠φ,f(c_2)≥a or if a_1<f(c_2)<a,f^2(c_2)>f(c_2),(resp.f has more than one fixed point,furthermore,if A≠φ and B\{b}≠φ,f(c_1)≤b or if b<f(c_2)<b_1,f^2(c_1)<f(c_1)). (3)If f(c_1)>c_2 and f(c_2)<c_1,then f has a single fixed point,a single period two orbit lying in I\(u,v)but no points of period greater than two,where u,v ∈[c_1,c_2] such that f(u)=c_2 and f(v)=c_1.
