We prove that every topologically stable homeomorphism with global attractor of $\mathbb {R}^n$ is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems, 1978, pp. 231–244).

中文翻译：

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具有全局吸引子的同胚的拓扑稳定性

我们证明每个具有全局吸引子$\mathbb {R}^n$的拓扑稳定同胚在其全局吸引子上都是拓扑稳定的。反之则不然。另一方面，如果局部紧度量空间的具有全局吸引子的同胚是可扩展的并且具有遮蔽性质，则它是拓扑稳定的。这扩展了沃尔特斯稳定性定理（沃尔特斯，《论伪轨道追踪特性及其与稳定性的关系。动态系统中吸引子的结构》，1978 年，第 231-244 页）。