Some characteristics of mean sensitivity and Banach mean sensitivity using Furstenberg families and inverse limit dynamical systems are obtained. The iterated invariance of mean sensitivity and Banach mean sensitivity are proved. Applying these results, the notion of mean sensitivity and Banach mean sensitivity is extended to uniform spaces. It is proved that a point-transitive dynamical system in a Hausdorff uniform space is either almost (Banach) mean equicontinuous or (Banach) mean sensitive.

Let R r 0 , R r 1 : S 1 − → S 1 be rotations on the unit circle S 1 and define f : Σ 2 × S 1 − → Σ 2 × S 1 as f (x, t) = (σ (x), R r x 1 (t)), for x = x 1 x 2 • • • ∈ Σ 2 := {0, 1} N , t ∈ S 1 , where σ : Σ 2 − → Σ 2 is the shift, and r 0 and r 1 are rotational angles. It is first proved that the system (Σ 2 × S 1 , f) exhibits maximal distributional chaos for any r 0 , r 1 ∈ R (no assumption of r 0 , r 1 ∈ R \ Q), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91-99, 2014). It is also obtained that (Σ 2 × S 1 , f) is cofinitely sensitive and (M 1 ,M 1)-sensitive and that (Σ 2 × S 1 , f) is densely chaotic if and only if r 1-r 0 ∈ R \ Q.

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Nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms. This paper obtains an equivalent characterization for the existence of idempotent nullnorms with the zero element a on any bounded lattice containing two distinct elements incomparable with a. Furthermore, some basic properties for the bounded lattice containing two distinct element incomparable with a are presented.