A Note on Stronger Forms of Sensitivity for Non-Autonomous Dynamical Systems on Uniform Spaces

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 $\mathcal{N}$-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 $(S,\mathcal{U})$, the system $(S,f_k\circ \cdots \circ f_1)$ exhibits $\mathcal{N}$-sensitivity (or strongly multi-sensitivity) if and only if the system $(S,f_{1,\infty })$ displays $\mathcal{N}$-sensitivity (or strongly multi-sensitivity). (2) Consider a finitely generated family of surjective maps on uniform space $(S,\mathcal{U})$. If the system $(S,f_{1,\infty })$ is $\mathcal{N}$-sensitive, then the system $(S,f_{k,\infty })$ is also $\mathcal{N}$-sensitive. When the family $f_{1,\infty }$ is feebly open, the converse statement holds true as well. (3) Within a finitely generated family on uniform space $(S,\mathcal{U})$, $\mathcal{N}$-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 $\mathcal{N}$-sensitivity.