Distributionally ‎chaotic dynamical systems in uniform spaces

Ahmadi, Seyyed Alireza

doi:10.22103/jmmr.2026.26798.1943

Distributionally ‎chaotic dynamical systems in uniform spaces

Articles in Press

Document Type : Research Paper

Author

Seyyed Alireza Ahmadi

Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

10.22103/jmmr.2026.26798.1943

Abstract

‎This study expands on the connections among various degrees of chaos within dynamical systems defined on uniform spaces‎, ‎concentrating on subsets where the trajectories of multiple points display irregular‎, ‎or‎ ‎"scrambled,"‎ ‎behavior as time progresses‎. ‎Our investigation centers on two primary frameworks‎: ‎distributional chaos and Li–Yorke chaos‎. ‎We initially address the circumstances under which the presence of a chaotic set containing $n$ points guarantees the existence of a chaotic set with $m$ points‎, ‎where $2 \leq m < n$‎. ‎We prove that under a condition called DC‎, ‎a distributionally topological $2$-scrambled set is a distributionally topological $n$-scrambled‎. ‎Although the forward direction is typically valid‎, ‎establishing the reverse implication necessitates additional hypotheses‎, ‎which we delineate.

Keywords

scrambled set‎

‎transitive map‎

‎distributional chaos‎

‎Li-yourk chaos‎

‎uniform space

Subjects

Differential Equations and Dynamical Systems

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