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<p>Devaney defines a function as chaotic if it satisfies the following three conditions: transitivity, having a dense set of periodic points, and sensitive dependence on initial conditions. In Banks et al. (Amer Math Monthly 99(4), 332–334, 1992), it was demonstrated that the first two conditions imply the third. This result was generalized in Akin et al. (When is a transitive map chaotic? de Gruyter, Berlin, 1996) by replacing the density of periodic points with the density of minimal points. The result was further generalized in Glasner (Am Math Soc, Providence, 2003) for group actions, in Kontorovich and Megrelishvili (Semigroup Forum 76, 133–141, 2008) Kontorovich and Megrelishvili (Semigroup Forum 76, 133–141, 2008) for Csemigroups actions, and in Dai (J Differ Equ 258(8), 2794–2805, 2015) for acontinuous semi-flow with X being a Polish space. Subsequently, in Iglesias and Portela (J Dyn Control Syst 28(4), 945–949, 2022) and Iglesias and Portela (Semigroup Forum 98(2), 261–270, 2019), it was generalized for compact spaces and for non-compact spaces in Zhukova (Chaotic Dyn 29(1), 174–189, 2024). The objective of this work is to generalize the result in Zhukova (Chaotic Dyn 29(1), 174–189, 2024), providing a simple proof. </p>

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Sensitive Actions on Non-compact Spaces

  • 期刊论文

  • 发表日期：

    2026年7月2日

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  • Jorge Iglesias,
  • Aldo Portela

摘要

Devaney defines a function as chaotic if it satisfies the following three conditions: transitivity, having a dense set of periodic points, and sensitive dependence on initial conditions. In Banks et al. (Amer Math Monthly 99(4), 332–334, 1992), it was demonstrated that the first two conditions imply the third. This result was generalized in Akin et al. (When is a transitive map chaotic? de Gruyter, Berlin, 1996) by replacing the density of periodic points with the density of minimal points. The result was further generalized in Glasner (Am Math Soc, Providence, 2003) for group actions, in Kontorovich and Megrelishvili (Semigroup Forum 76, 133–141, 2008) Kontorovich and Megrelishvili (Semigroup Forum 76, 133–141, 2008) for Csemigroups actions, and in Dai (J Differ Equ 258(8), 2794–2805, 2015) for acontinuous semi-flow with X being a Polish space. Subsequently, in Iglesias and Portela (J Dyn Control Syst 28(4), 945–949, 2022) and Iglesias and Portela (Semigroup Forum 98(2), 261–270, 2019), it was generalized for compact spaces and for non-compact spaces in Zhukova (Chaotic Dyn 29(1), 174–189, 2024). The objective of this work is to generalize the result in Zhukova (Chaotic Dyn 29(1), 174–189, 2024), providing a simple proof.