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We consider positive entropy $G$-systems for certain countable, discrete,infinite left-orderable amenable groups $G$. By undertaking local analysis, theexistence of asymptotic pairs and chaotic sets will be studied in connectingwith the stable sets. Examples are given for the case of integer latticegroups, the Heisenberg group, and the groups of integral unipotent uppertriangular matrices.

Asymptotic pairs, stable sets and chaos in positive entropy systems