# Li–Yorke sensitivity for semigroup actions

### Abstract

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.

Published

25.05.2013

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 65, no. 5, May 2013, pp. 681–688, http://umj.imath.kiev.ua/index.php/umj/article/view/2451.

Issue

Section

Research articles