Abstract In this paper, we investigate dynamical properties of monoid actions on symbolic systems. First, we establish mixing properties and an entropy formula for such actions. To describe chaotic behavior, we introduce and distinguish several types of Li–Yorke chaos. Our main result shows that positive entropy is equivalent to locally Li–Yorke chaos, a notion that strengthens the classical definition of Li–Yorke chaos.
Ban et al. (2026) studied this question.