2017
Том 69
№ 9

All Issues

Li–Yorke sensitivity for semigroup actions

Rybak O. V.

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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.

English version (Springer): Ukrainian Mathematical Journal 65 (2013), no. 5, pp 752-759.

Citation Example: Rybak O. V. Li–Yorke sensitivity for semigroup actions // Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 681–688.

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