以下为句子列表：
英文: The present paper presents the chain recurrent set with the expansive homeomorphism and shadowing properties and this connection for the mistake in literature [1] and its proof.
中文: 摘要给出具有跟踪性可扩同胚的链回归集的几个性质，指出以往文献中证明链回归集无环性时的错误之处，并给出严格的证明。
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