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<p>In this paper, we introduce the notions of topological expansiveness, the topological shadowing property and topological stability for Borel measures with respect to homeomorphisms on metric spaces. We prove that on a locally compact metric space every topologically expansive measure with the topological shadowing property is topologically stable. This represents a measurable version on locally compact case of the classical Walters’s stability theorem. Moreover, we extend the Smale’s spectral decomposition theorem for topologically strong measure expansive homeomorphisms on locally compact spaces. Some properties of topologically stable measures are also considered.</p>

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Topological stability and topological shadowing property for topologically expansive measures on metric spaces

  • 期刊论文

  • 发表日期：

    2025年2月18日

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  • Yutong Huo,
  • Yinong Yang

摘要

In this paper, we introduce the notions of topological expansiveness, the topological shadowing property and topological stability for Borel measures with respect to homeomorphisms on metric spaces. We prove that on a locally compact metric space every topologically expansive measure with the topological shadowing property is topologically stable. This represents a measurable version on locally compact case of the classical Walters’s stability theorem. Moreover, we extend the Smale’s spectral decomposition theorem for topologically strong measure expansive homeomorphisms on locally compact spaces. Some properties of topologically stable measures are also considered.