Abstract Let f : X → X be a continuous map on a compact metric space X and let α f , ω f and I C T f denote the set of α-limit sets, ω-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map f has shadowing then every element of I C T f can be approximated (to any prescribed accuracy) by both the α-limit set and the ω-limit set of a full-trajectory. Furthermore, if f is additionally expansive then every element of I C T f is equal to both the α-limit set and the ω-limit set of a full-trajectory. In particular this means that shadowing guarantees that α f ‾ = ω f ‾ = I C T f (where the closures are taken with respect to the Hausdorff topology on the space of compact sets), whilst the addition of expansivity entails α f = ω f = I C T f . We progress by introducing novel variants of shadowing which we use to characterise both maps for which α f ‾ = I C T f and maps for which α f = I C T f .

中文翻译：

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阴影、内部链传递性和 α 极限集

摘要 令 f : X → X 是紧致度量空间 X 上的连续映射，令 α f 、ω f 和 ICT f 分别表示 α-极限集、ω-极限集和非空闭内部链传递集的集合。我们表明，如果映射 f 有阴影，那么 ICT f 的每个元素都可以通过全轨迹的 α 极限集和 ω 极限集来近似（达到任何规定的精度）。此外，如果 f 是额外膨胀的，那么 ICT f 的每个元素都等于完整轨迹的 α 极限集和 ω 极限集。特别地，这意味着阴影保证 α f ‾ = ω f ‾ = ICT f（其中闭包是针对紧集空间上的 Hausdorff 拓扑进行的），而扩展性的增加需要 α f = ω f =信息通信技术 f .