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For a topological dynamical system (X, T), l ∈ N and x ∈ X , let N l (X) and L x l (X) be the orbit closures of the diagonal point (x , ... , x) (l times) under the actions G l and τ l respectively, where G l is generated by T × ... × T (l times) and τ l = T × ... × T l . In this paper, we show that for a minimal system (X, T) and d , l ∈ N , the maximal d-step pro-nilfactor of (N l (X) , G l) is (N l (X d) , G l) , where X d is the d-step pronilfactor of (X, T). Meanwhile, when (X, T) is a minimal nilsystem, we also calculate the pro-nilfactors of the system (L x l (X) , τ l) for almost every x w.r.t. the Haar measure. In particular, there exists a minimal 2-step nilsystem (Y, T) and a countable subset Ω of Y such that for every y ∈ Y \ Ω the maximal equicontinuous factor of (L y 2 (Y) , τ 2) is not (L π 1 (y) 2 (Y 1) , τ 2) , where Y 1 is the maximal equicontinuous factor of (Y, T) and π 1 : Y → Y 1 is the factor map. [ABSTRACT FROM AUTHOR]

In this note, it is shown that several results concerning mean equicontinuity proved before for minimal systems are actually held for general topological dynamical systems. Particularly, it turns out that a dynamical system is mean equicontinuous if and only if it is equicontinuous in the mean if and only if it is Banach (or Weyl) mean equicontinuous if and only if its regionally proximal relation is equal to the Banach proximal relation. Meanwhile, a relation is introduced such that the smallest closed invariant equivalence relation containing this relation induces the maximal mean equicontinuous factor for any system. [ABSTRACT FROM AUTHOR]