圆周自映射的v极限集

ν-LIMIT SET FOR MAPS OF THE CIRCLE

设 f:s^1→s^1为连续映射。f 的回归点集和非游荡集分别记为 R 和Ω.xes^1,令v(x)=ω(x)∩α(x),其中ω(x)(α(x)为 x 的ω-(α-)极限集.令Γ=(?)v(x),若 y(?)s^1,记∧(y)=(?)ω(x).我们证明了:(1)Γ=∧(Ω)=∧(∧)=∧(Γ);(2)Ω-Γ是 s^1中无处稠密的可数集;(3)若以 x 为端点的每个开弧至少包含某个轨道中的的两点,则 x∈Γ;(4)若Γ-R≠φ,则Γ-R 为不可数集;(5)如(?)-R≠φ,则(?)-R 为无限集;(6)Γ=R 当且仅当(?)^(+)∩(?)^(-)=R.其中(?)^(+)((?)^(-))表示 R 的右(左)闭包。

We study the dynamics of continuous maps of the circle.Denote Rand Ω the set of recurrent points,and set of nonwandering points,respec-tively.A point x∈S^1,we denote the set of ω-limit points and α-limit pointsof x by ω(x) and α(x).Let v(x)=ω(x)∩α(x),Γ=(?) v(x).For any subsetY of S^1,denote ∧(Y)=(?)ω(x).We show that(1)Γ=∧(Ω)=∧(∧)=∧(Γ)(2)Ω-Γ is countable and (?)nowhere dense in S′.(3)A point x lies in Γ ifevery open interval with endpoint x contains at least two points of someorbit.(4)If Γ-R is not empty then it is uncountable.(5)If R-(?) is not em-pty then it is infinite.(6)Γ=R if and only if (?)=R.