无马蹄的圆周自映射

CONTINUOUS SELF-MAPS OF THE CIRCLE WITHOUT HORSESHOES

对于圆周连续自映射f,我们证明了以下条件是彼此等价的:(1)f无马蹄。(2)f的回归点集的闭包中的非回归点的集合为可数集。(3)任何一点的ω-极限集只含有唯一的极小集。(4)任何一点的ω-极限点或是f的一个周期轨道或者不包含f的周期轨道。(5)任一点的ω-极限点的ω-极限点是回归点。(6)任一点的ω-极限点的ω-极限点是几乎周期点。(7)每一个非游荡点的ω-极限集是极小集。

For continuous maps f of the circle to itself, We show the following conditions are equivalent(1) f has no horseshoe.(2) —R is countable, where R denotes the set of recurrent points.(3) ω-limit set of every point contains only unique minimal set.(4) ω-limit set of every point is a periodic orbit of f or contains no periodic orbit of f.(5) every ω-limit point of an ω-limit point is recurrent point.(6) every ω-limit point of an ω-limit point is almost periodic point.(7) ω-limit set of every nonwandering point is a minimal set.