On Li-Yorke pairs

The Li-Yorke definition of chaos proved its value for interval maps. In this paper 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.