sin(1/x)连续统上每个点都为链回归点的映射

Maps of the sin(1/x) Continuum with Every Point Chain Recurrent

摘要: 设S为sin(1/x)连续统，f:S→S为连续自映射，其中S=L₁∪L₂ ，L₁={(x,y)∈R²|x=0,-1≤y≤1} ，L₂={(x,y)∈R²|sin(1/x),0≤x≤1} 。本文指出：如果f为逐点链回归映射，那么，若Fin(f) 连通，则f为恒等映射；若Fin(f) 不连通，则当Fin(f₁) 或者Fin(f₂) 非退化不连通时， f含湍流，当Fin(f₁)=L₁ ， Fin(f₂)=a,a∈L₂且(L_2-{a})∩P(f₂)=φ 时， f不含湍流。

Abstract: Let S be sin(1/x) continuum and f:S→S is a continuous map, where S=L₁∪L₂ , L₁={(x,y)∈R²|x=0,-1≤y≤1} , L₂={(x,y)∈R²|sin(1/x),0≤x≤1} . It is showed that if f is pointwise chain recurrent, then if Fin(f) is connected, f is identify; if Fin(f) is disconnected, then f is turbulent while Fin(f₁) or Fin(f₂) is nondegenerate disconnected; f  is not turbulent while Fin(f₁)=L₁ , Fin(f₂)=a,a∈L₂ and (L_2-{a})∩P(f₂)=φ.