可降映射的一些动力学性质

Some Dynamical Properties of Descendable mappings.

讨论了可降映射的性质,得到了fi(i=1,2,…,k)为f的下降组(即f为可降映射)的等价条件,并给出一个简洁的证明,也得到了两个可降映射的复合和笛卡尔乘积是可降映射。设f∈0∏ki=1Ii,∏ki=iIi是可降映射,fi(i=1,2,…,k)是f的下降组,证明了:若f有m-周期点,且m n,则fi必有n-周期点,i=1,2,…,k;设m为f的一个周期,则对每个满足m n的正整数n,f有n-周期点当且仅当对每个fi,i=1,2,…,k,存在fi的周期mi,使得正整数t满足mi t时,fi就有t-周期点,其中[m1,m2,…,mk]=m.

In this paper, the properties of descendable mappings were discussed. An equivalent condition of f with the descendable system fi（i=1,2, ... ,k） was obtained. And a simple proof of the lemma 2 was given. It was also obtained that compotition and Cartesian product of two dekscendable mappings are descendable mappings. Let f ∈ C°（ kⅡi=1Ii,kⅡi=iIi） be a descendable mapping. It was proved that （1）If f has a periodic point of period m and m△ n, then fi also has a periodic point of period n, i= 1,2,……k. （2）Let m be a period of f, then for every positive integer n with m n, f has a periodic point of period n if and only if fi has a periodic point of period mi （i=1,2,…, k） such that[m1 ,m2 ,… ,mk]=m and if t is a positive integer and mi△t, ,then fi has a periodic point of period t（i=1,2, ... ,k）.