区间上平顶单峰扩张自映射的周期轨道

PERIODIC ORBITS OF LEVEL TOP UNIMODAL EXPANDING SELF MAPS ON THE INTERVAL

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)) .