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<p>We investigate the dynamics of periodic non-autonomous discrete dynamical systems on uniform spaces and topological spaces, focusing on the extension of the classical Auslander–Yorke dichotomy to these settings. We prove various dichotomy theorems in the uniform-space framework, showing that a minimal periodic non-autonomous system is either sensitive or equicontinuous, and prove some more refined versions involving syndetic equicontinuity and thick sensitivity and eventual sensitivity versus equicontinuity on compact uniform spaces. We further introduce topological analogues like topological equicontinuity, Hausdorff sensitivity, and their syndetic and multi-sensitive variants and prove corresponding Auslander–Yorke–type dichotomies on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(T_3\)</EquationSource> </InlineEquation> spaces.</p>

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Auslander–Yorke Dichotomy and Its Generalizations for Non-autonomous Dynamical Systems

  • 期刊论文

  • 发表日期：

    2026年5月9日

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  • Saksham Malik,
  • Mohammad Salman,
  • Ruchi Das

摘要

We investigate the dynamics of periodic non-autonomous discrete dynamical systems on uniform spaces and topological spaces, focusing on the extension of the classical Auslander–Yorke dichotomy to these settings. We prove various dichotomy theorems in the uniform-space framework, showing that a minimal periodic non-autonomous system is either sensitive or equicontinuous, and prove some more refined versions involving syndetic equicontinuity and thick sensitivity and eventual sensitivity versus equicontinuity on compact uniform spaces. We further introduce topological analogues like topological equicontinuity, Hausdorff sensitivity, and their syndetic and multi-sensitive variants and prove corresponding Auslander–Yorke–type dichotomies on \(T_3\) spaces.