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<p>First we show that the eventual shadowing property, chain transitivity, and chain recurrence for homeomorphisms are not invariant under topological conjugacy. Afterwards, we introduce the concepts of <i>topological eventual shadowing property</i>, <i>topological chain transitivity</i>, and <i>topological chain recurrence</i>. We show that these redefined properties are invariant under topological conjugacy and reduce to their classical counterparts in compact spaces. By extending the classical shadowing property, we show how the topological eventual shadowing property applies to the spectral decomposition theorem, generalizing Smale’s framework for expansive homeomorphisms on compact spaces. Additionally, we analyze relationships between chain recurrence, chain transitivity, and the newly defined topological chain properties in both compact and noncompact settings, providing fresh insights into the structure and dynamics of chain-recurrent sets and their role in system mixing and classification.</p>

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Eventual Shadowing and Chain Recurrence: Extending Spectral Decomposition in Noncompact Dynamics

  • 期刊论文

  • 发表日期：

    2026年3月14日

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  • Jiahui Liu,
  • Meihua Dong

摘要

First we show that the eventual shadowing property, chain transitivity, and chain recurrence for homeomorphisms are not invariant under topological conjugacy. Afterwards, we introduce the concepts of topological eventual shadowing property, topological chain transitivity, and topological chain recurrence. We show that these redefined properties are invariant under topological conjugacy and reduce to their classical counterparts in compact spaces. By extending the classical shadowing property, we show how the topological eventual shadowing property applies to the spectral decomposition theorem, generalizing Smale’s framework for expansive homeomorphisms on compact spaces. Additionally, we analyze relationships between chain recurrence, chain transitivity, and the newly defined topological chain properties in both compact and noncompact settings, providing fresh insights into the structure and dynamics of chain-recurrent sets and their role in system mixing and classification.