Abstract:Let be a continuous function on a compact metric space. We show that shadowing is equivalent to backwards shadowing and two-sided shadowing when the map is onto. Using this we go on to show that, for expansive surjective maps the properties shadowing, two-sided shadowing, s-limit shadowing, and two-sided s-limit shadowing are equivalent. We show that is positively expansive and has shadowing if and only if it has unique shadowing (i.e., each pseudo-orbit is shadowed by a unique point), extending a result implicit in Walter's proof that positively expansive maps with shadowing are topologically stable. We use the aforementioned result on two-sided shadowing to find an equivalent characterisation of shadowing and expansivity and extend these results to the notion of -expansivity due to Morales.

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中文翻译：

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扩展性和独特的阴影

摘要：让我们成为紧度量空间上的连续函数。我们展示了在贴图时，阴影等效于向后阴影和双面阴影。使用此方法，我们继续表明，对于可扩展的射影图，阴影，两侧阴影，s限制阴影和两侧s限制阴影是等效的。我们证明，当且仅当它具有唯一的阴影（即，每个伪轨道都被一个唯一的点阴影）时，它才是正膨胀的，并且具有阴影，从而扩展了Walter证明含阴影的正膨胀图在拓扑上稳定的隐式结果。我们将上述结果用于双面阴影，以找到阴影和膨胀率的等效特征，并将这些结果扩展到 -由于莫拉莱斯的扩张。

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