We prove that every finite-expansive homeomorphism with the shadowing property has a kind of stability. This stability will be good enough to imply both the shadowing property and the denseness of periodic points in the chain recurrent set. Next we analyze the N-shadowing property which is really stronger than the multishadowing property in Cherkashin and Kryzhevich (Topol Methods Nonlinear Anal 50(1): 125–150, 2017). We show that an equicontinuous homeomorphism has the N-shadowing property for some positive integer N if and only if it has the shadowing property.

中文翻译：

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有限膨胀和N-阴影

我们证明，具有遮蔽性质的每个有限扩张同胚都具有一种稳定性。这种稳定性将足以暗示链递归集中的阴影特性和周期点的密度。接下来，我们分析N阴影属性，该属性确实比Cherkashin和Kryzhevich中的多阴影属性强（Topol Methods Nonlinear Anal 50（1）：125–150，2017）。我们证明了一个同等连续同胚有ñ对于一些正整数-shadowing财产ñ当且仅当它具有跟踪性。