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  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 , U) , the system  (S , f k ∘ ⋯ ∘ f 1)  exhibits  N -sensitivity (or strongly multi-sensitivity) if and only if the system  (S , f 1 , ∞)  displays  N -sensitivity (or strongly multi-sensitivity). (2) Consider a finitely generated family of surjective maps on uniform space  (S , U) . If the system  (S , f 1 , ∞)  is  N -sensitive, then the system  (S , f k , ∞)  is also  N -sensitive. When the family  f 1 , ∞  is feebly open, the converse statement holds true as well. (3) Within a finitely generated family on uniform space  (S , U) ,  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  N -sensitivity. [ABSTRACT FROM AUTHOR]