摘要
设t(0<t<1)是一个常数,n≥3是奇数,m≥0及k≥1是整数,P0(x)=x-1,Pi(x)=(x2i-1-1)Pi-1(x)(i≥1),rmn(t)及rk(t)分别是方程Pm(x)(x2mn-2x2m(n-2)-1)-t(x2mn-1)(x2m+1)=0及Pk-1(x)-t(x2k-1+1)=0在(1,+∞)上的唯一实根,f是闭区间I=[0,1]上的峰顶区间长度为t的平顶单峰扩张自映射.本文证明了,若f的扩张常数λ≥rmn(t)(或>rk(t)),,则f有2mn(或2k)周期点.此外,本文还指出,当1<λ<rmn(t)(或≤rk(t)时。
Let t(0<t<1) is a constant,and p 0(x)=x-1,p i(x)=(x 2 i-1 -1)p i-1 (x)(i≥1) ,denote by r mn (t) and r k(t) the unique real roots of the equations P m(x)(x 2 mn -2x 2 m -1)-t(x 2 mn -1)(x 2 m +1)=0 and P k-1 (x)-t(x 2 k-1 +1)=0 on (1,+∞) ,respectively,where n(≥3) is odd number, m(≥0) and k(≥1) intergers.Assume f is a level top unimodal expanding self map which length of top interval is t on closed interval I= .In this paper.it is proved that f has a periodic point of period 2 mn ( or 2 k) if f has an expanding constant λ≥r mn (t) ( or >r k(t)) .Besides.it show that there exits a level top unimodal expanding self map which has an expanding constant λ and length of top interval t but no periodic point of period 2 mn ( or 2 k) on I if 1<λ<r mn (t) ( or ≤r k(t)) .
出处
《数学杂志》
CSCD
北大核心
1996年第3期312-320,共9页
Journal of Mathematics
关键词
单峰扩张自映射
周期点
周期轨道
平顶单峰映射
level top unimodal expanding self map expanding constant periodic point.