$\Delta$-Weakly Mixing Subset In Positive Entropy Actions Of A Nilpotent Group

Abstract

The notion of $\Delta$-weakly mixing subsets is introduced for countable torsion-free discrete group actions. It is shown that for a finitely generated torsion-free discrete nilpotent group action, positive topological entropy implies the existence of $\Delta$-weakly mixing subsets, and while there exists a finitely generated torsion-free discrete solvable group action which has positive topological entropy but without any $\Delta$-weakly mixing subsets.