Li-Yorke and Devaney chaotic uniform dynamical systems amongst weighted shifts

TL;DR

This paper establishes precise conditions under which weighted generalized shift dynamical systems over finite fields exhibit Li-Yorke and Devaney chaos, enhancing understanding of chaos in discrete weighted shifts.

Contribution

It provides necessary and sufficient criteria for chaos in weighted generalized shift systems over finite fields, a novel characterization in this context.

Findings

Criteria for Li-Yorke chaos established

Criteria for Devaney chaos established

Conditions depend on weights and shift functions

Abstract

In this paper, for finite discrete field $F$, nonempty set $\Gamma$, weight vector $\mathfrak{w}=({\mathfrak w}_\alpha)_{\alpha\in\Gamma}\in F^\Gamma$ and weighted generalized shift $\sigma_{\varphi,{\mathfrak w}}:F^\Gamma\to F^\Gamma$, we find necessary and sufficient conditions for uniform dynamical system $(F^\Gamma,\sigma_{\varphi,{\mathfrak w}})$ to be Li--Yorke chaotic. Next we find necessary and sufficient conditions for $(F^\Gamma,\sigma_{\varphi,{\mathfrak w}})$ to be Devaney chaotic.