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

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 ${\sigma }_{\phi ,\mathfrak{w}}:F^{\Gamma }\to F^{\Gamma }$, we find necessary and sufficient conditions for uniform dynamical system $(F^{\Gamma },{\sigma }_{\phi ,\mathfrak{w}})$ to be Li--Yorke chaotic. Next we find necessary and sufficient conditions for $(F^{\Gamma },{\sigma }_{\phi ,\mathfrak{w}})$ to be Devaney chaotic.

15 pages