交换环上幂映射的周期轨道分支的对称性

设Ｒ是个交换环，带离散拓扑，是由ｆ（ｒ）＝ｒ￣ｎ（任ｒ∈Ｒ）定义的幂映射．又设ｘ及ｙ均是ｆ的周期点，其周期分别是ｋ及ｌ．记称Ｗ＿ｘ为ｆ的含ｘ的周期轨道分支，本文证明：（１）Ｗ＿ｘ在ｆ之下具有循环对称性，即存在着周期为ｋ的周期映射ｈ＿ｘ：Ｗ＿ｘ→Ｗ＿ｘ使得ｆｈ＿ｘ＝ｈ＿ｘｆ｜Ｗ＿ｘ且ｈ＿ｘ（ｘ）＝ｆ（ｘ）；（２）当ｌ是ｋ的因数且存在ｕ∈Ｒ使得ｙ＝ｕｘ时，　存在着映射ξ＿ｕ：Ｗ＿ｘ→Ｗ＿ｙ满足；（ｉｉｉ）若还存在着ｖ∈Ｒ使得ｘ＋ｖｙ，且ｌ＝ｋ，则此ξ＿ｖ与ξ＿ｖ互逆．

The Symmetry of Components Containing Periodic Orbits of Power Maps on Com-mutative Rings

Let R be a commutative ring and f:R→R a power map defined by f(r)=r￣nLet x and y be two periodic points of f witli periods k and l respectively. Setand.In this paper we prove:(1)There exists a periodic map h_x:W_x→W_x of period k such that fh_x=h_xf|W_x and h_x(x)=f(x);(2)If l is a factor of kand y = ux for some u ∈ R, then there exsits a map ξ_u∶W_x→W_y satisfying (a)and(c)If l=k and x=vy also holds for some v ∈ R,then the maps ξ_u andξ_v∶W_y→W_x are mutually inverse.