A Note on Stronger Forms of Sensitivity for Non-Autonomous Dynamical Systems on Uniform Spaces - Lixin Jiao; Heyong Wang; Lidong Wang; Nan Wang (Entropy, 2024)
This paper introduces the notion of multi-sensitivity with respect to a vector within the context of non-autonomous dynamical systems on uniform spaces and provides insightful results regarding <semantics>N</semantics>-sensitivity and strongly multi-sensitivity, along with their behaviors under various conditions. The main results established are as follows: (1) For a k-periodic nonautonomous dynamical system on a Hausdorff uniform space <semantics>(S,U)</semantics>, the system <semantics>(S,fk∘⋯∘f1)</semantics> exhibits <semantics>N</semantics>-sensitivity (or strongly multi-sensitivity) if and only if the system <semantics>(S,f1,∞)</semantics> displays <semantics>N</semantics>-sensitivity (or strongly multi-sensitivity). (2) Consider a finitely generated family of surjective maps on uniform space <semantics>(S,U)</semantics>. If the system <semantics>(S,f1,∞)</semantics> is <semantics>N</semantics>-sensitive, then the system <semantics>(S,fk,∞)</semantics> is also <semantics>N</semantics>-sensitive. When the family <semantics>f1,∞</semantics> is feebly open, the converse statement holds true as well. (3) Within a finitely generated family on uniform space <semantics>(S,U)</semantics>, <semantics>N</semantics>-sensitivity (and strongly multi-sensitivity) persists under iteration. (4) We present a sufficient condition under which an nonautonomous dynamical system on infinite Hausdorff uniform space demonstrates <semantics>N</semantics>-sensitivity.