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On Li-Yorke pairs. (English) Zbl 1059.37006
Summary: The Li-Yorke definition of chaos proved its value for interval maps. Here, it is considered in the setting of general topological dynamics. We adopt two opposite points of view. On the one hand, sufficient conditions for Li-Yorke chaos in a topological dynamical system are given.We solve a long-standing open question by proving that positive entropy implies Li-Yorke chaos. On the other hand, properties of dynamical systems without Li-Yorke pairs are investigated; in addition to having entropy 0, they are minimal when transitive, and the property is stable under factor maps, arbitrary products and inverse limits. Finally, it is proved that minimal systems without Li-Yorke pairs are disjoint from scattering systems.

MSC:
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37A25 Ergodicity, mixing, rates of mixing
37B40 Topological entropy
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