Том 71
№ 11

All Issues

Rybak O. V.

Articles: 1
Article (Ukrainian)

Li–Yorke sensitivity for semigroup actions

Rybak O. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 681–688

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.