The authors deal with various forms of transitivity and related notions of mixing and obtain for them dichotomy results. They prove that the map \(f\) on a compact metric space \((X,d)\) is called chain mixing, strong chain mixing, or vague mixing if the product function \(f\times f\) on the product metric space \((X \times X, d \times d)\) is chain transitive, strong chain transitive, or vague transitive. By using the barrier functions of A. Fathi and P. Pageault [Ergodic Theory Dyn. Syst. 35, No. 4, 1187–1207 (2015; Zbl 1343.37012)], the authors obtain dichotomy results for these notions of transitivity and mixing. In particular, the authors prove that if \(f\) is a minimal homeomorphism on a compact metric space \((X, d)\), then \(f\) is weak mixing if and only if it is vague mixing.