Li–Yorke sensitivity for semigroup actions

  • O. V. Rybak

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 xX and any open neighborhood U of this point, one can find a point yU for which the following conditions are satisfied:
(i) d(g(x), g(y)) > ε for infinitely many gG,
(ii) for any δ > 0; there exists hG 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
RybakO. V. “Li–Yorke Sensitivity for Semigroup Actions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, no. 5, May 2013, pp. 681–688, http://umj.imath.kiev.ua/index.php/umj/article/view/2451.
Section
Research articles