Li–Yorke sensitivity for semigroup actions

We introduce and study the concept of Li–Yorke sensitivity for semigroup actions (dynamical systems of the form (X, G), where X is a metric space and G is a semigroup of continuous mappings of this space onto itself). A system (X, G) is called Li–Yorke sensitive if there exists positive ε such that, for any point x ∈ X and any open neighborhood U of this point, one can find a point y ∈ U for which the following conditions are satisfied: (i) d(g(x), g(y)) > ε for infinitely many g ∈ G, (ii) for any δ > 0; there exists h ∈ G satisfying the condition d(h(x), h(y)) < δ. In particular, it is shown that a nontrivial topologically weakly mixing system (X, G) with a compact set X and an Abelian semigroup G is Li–Yorke sensitive.