交换环上含参数的幂映射的周期轨道分支的对称性

The Symmetry of Components Containing Periodic Orbits of Parametric Power Maps on Commutative Rings

设Ｒ是个交换环，带有离散拓扑，ｆｔ：Ｒ→Ｒ是由ｆｔ（ｘ）＝ｔｘｎ（任意ｘ∈Ｒ）定义的映射，ｎ≥２，ｔ∈Ｎ是参数。又设ｘ、ｙ是ｆｔ的周期点，其周期分别是ｋ及ｌ。记Ｗｘ＝∪∞ｉ＝０ｆ－ｉｔ（ｘ），Ｗｙ＝∪∞ｉ＝０ｆ－ｉｔ（ｙ），称Ｗｘ为含有ｘ的周期轨道分支。本文证明了，Ａ：Ｗｘ在ｆｔ之下具有循环对称性，即存在周期为ｋ的映射ｈｘ：Ｗｘ→Ｗｘ，使得ｆｔｈｘ＝ｈｘｆｔ｜Ｗｘ，且ｈｘ（ｘ）＝ｆｔ（ｘ）；Ｂ：当ｌ是ｋ的因数且存在ｕ∈Ｒ使得ｙ＝ｕｘ时，存在映射ζｕ：Ｗｘ→Ｗｙ满足①ｆｔζｕ＝ζｕｆｔ｜Ｗ；②ζｕｈｘ＝ｈｙζｕ；③若还存在ｖ∈Ｒ使得ｘ＝ｖｙ，且ｌ＝ｋ，则此ζｕ与ζｖ互为逆映射。

Let R be a commutative ring and f t : R→R a parametric power map defined by f t (x)=tx n( x∈R,n≥2 )t∈N a parametry. Let x and y be two periodic points of f t with periods k and l respectively. Set W x=∪∞i=0f -i t(x) and W y=∪∞i=0f -i t(y). It has been proved: (1) There exists a periodic map h x:W x→W x of period k such that f th x=h xf t|W x and h x(x)=f t(x);(2) If l is a factor of k and y=ux for some u∈R, then there exists a map ζ u:W x→W y satisfying (a) f tζ u=ζ uf t|W x,(b) ζ uh x=h yζ u,and (c) If l=k and x=vy also holds for some v∈R, then the maps ζ u and ξ: W y→W x are mutually inversed.