  We consider extensions of the notion of topological transitivity for a
dynamical system $(X,f)$. In addition to chain transitivity, we define strong
chain transitivity and vague transitivity. Associated with each there is a
notion of mixing, defined by transitivity of the product system $(X \times X, f
\times f)$. These extend the concept of weak mixing which is associated with
topological transitivity. Using the barrier functions of Fathi and Pageault, we
obtain for each of these extended notions a dichotomy result that a transitive
system of each type either satisfies the corresponding mixing condition or else
factors onto an appropriate type of equicontinuous minimal system. The
classical dichotomy result for minimal systems follows when it is shown that a
minimal system is weak mixing if and only if it is vague mixing.
