区间上三段单调扩张自映射的周期轨道

Periodic Orbits of Three-pieces Monotone Expanding Self-maps on the Interval

设ｎ３是奇数，ｍ０是整数，Ｓｎ及ｒｎ分别是方程ｘｎ－２ｘｎ－１－１＝０及ｘｎ＋２－３ｘｎ－ｘ２－１＝０的唯一正根，记ｔｎ０＝ｒｎ，ｔｎｉ＝ｓｎ（ｉ１）．又设ｆ及ｇ分别是闭区间Ｉ上的Ｎ型（即增－减－增型）及反Ｎ型（即减－增－减型）扩张自映射．本文证明了，若ｆ（或ｇ）的扩张常数或则ｆ（或ｇ）有２ｍ·ｎ周期点．此外，本文还指出，当或时，在［０，１］上存在着具有扩张常数λ却无２ｍ·ｎ周期轨道的Ｎ型（或反Ｎ型）扩张自映射．

Let be an odd number, integer, Sn and rn be the unique real roots ofthe equations xn-2xn-1 -1=0 and Xn+2-3Xn- X2-1=0 on(0,+∞) respectively,tno= rn,tni = sn ().Assume f and g be the N take of expanding self-map (increasedecrease-increase type)and anti-N type of expanding self-map(decrease-increase-decrease take)on closed interval I respectively. In this paper, it is proved that f(or g) has a periodic point of period 2m n if f (or g) has an expanding constant(or). Besides,it shows that there exists a N type of (or anti-N type of)expanding self-map which has an expanding constant λ but no periodic point of period 2m·n on [0, 1]if .