Sensitivity, Devaney’s chaos and Li–Yorke $$\varepsilon $$-chaos

Abstract

We consider the sensitivity from a topological point of view. We show that a continuous, topologically transitive and non-minimal action of a monoid S on an infinite $$T_4$$ topological space which admits a dense set of almost periodic points is sensitive. We also prove that a uniformly continuous, topologically transitive and non-minimal action of a monoid S on an infinite Hausdorff uniform space which admits a dense set of almost periodic points is thickly syndetically sensitive. We point out that if a continuous action of an Abelian group on a compact metric space is chaotic in the sense of Devaney and has a fixed point, then for every positive integer $$n\ge 2$$, it is Li–Yorke n–$$\varepsilon $$-chaotic for some $$\varepsilon >0$$. Moreover, we show that a continuous and transitive compact action of a semigroup S on a compact metric space is Li–Yorke sensitive and Li–Yorke $$\varepsilon $$-chaotic for some $$\varepsilon >0$$.