Weak mixing in general semiflows implies multi-sensitivity, but not thick sensitivity

Abstract

It was proved by Wang et al. [Wang, J. Yin, Q. Yan, J. Nonlinear Sci. Appl., 9 (2016), 989-997] that any weakly mixing semiflow on a compact metric space, whose all transition maps are surjective, is thickly sensitive. We consider what happens if we do not have the assumptions of compactness and surjectivity. We prove that even in that case any weakly mixing semiflow is multi-sensitive, and that, however, it does not have to be thickly sensitive.

A note about various types of sensitivity in general semiflows

“…This semiflow is weakly mixing since (X, f ) is weakly mixing. It was shown in [6] that (T, X) is not thickly sensitive.…”

“…Proof. We will follow [6]. If the diameter of X is infinite let D be any positive real number, otherwise let diam(X) = 12D > 0.…”

“…All notions that we mention, but not define, are standard and can be, for example found in [1,4,6]. Stronger forms of sensitivity were introduced in [8], and discussed, for example, in [3,5,8,9].…”