摘要: 设X为局部紧致度量空间，f：X→X是同胚映射。本文主要证明如下两个结论：1) 若对∀x∈X，ω(x)≠∅，K⊂X为紧致强不变集，且存在一个紧致邻域Q，使得Q\K包含不完整负轨道，则K是渐进稳定集；2) 若X的每一个有界闭集都是紧致的，且K⊂X为吸引子，则
是渐进稳定的，其中
与K有相同的吸引域。更多的，K是渐近稳定的当且仅当
=K 。结论1)，2)分别是对文献[1]和[2]中的结论进行了进一步推广。
Abstract:
Let X be a local compact metric space, f：X→X be a homeomorphism. This article mainly proves the following two conclusions: 1) Assume that
K⊂X is a compact strongly invariant set and there exists a compact neighborhood K⊃X , such that
Q\K contains no complete negative trajectory for every x∈X and
ω(x) is nonempty set, then K is asymptotically stable. 2) If each bounded closed set is a compact set for X and let
K⊂X be an attractor. Then
is asymptotically stable with the same basin of attraction that K. Moreover K is asymptotically stable if only if
=K .