A note on multivariate diam-mean equicontinuity and frequent stability

Let ( X , G ) be a topological dynamical system, given by the action of a countable discrete infinite group on a compact metric space X . We prove that every minimal system ( X , G ) is either diam-mean m -equicontinuous or diam-mean m -sensitive. We further introduce the notion of strong m -spreading and show that every minimal system ( X , G ) is either frequently m -stable or strongly m -spreading. Further, when G is abelian (or, more generally, (virtually) nilpotent), then the following statements are equivalent: (1) ( X , G ) is a regular m -to-one extension of its maximal equicontinuous factor; (2) ( X , G ) is diam-mean ( m + 1 ) -equicontinuous, and not diam-mean m -equicontinuous; (3) ( X , G ) is not diam-mean ( m + 1