  Recently, two stronger versions of dynamical properties have been introduced
and investigated: strong topological transitivity, which is a stronger version
of the topological transitivity property, and hypermixing, which is a stronger
version of the mixing property. We continue the investigation of these notions
with two main results. First, we show there are dynamical systems which are
strongly topologically transitive but not weakly mixing. We then show that on
$\ell^p$ or $c_0$, there is a weighted backward shift which is strongly
topologically transitive but not mixing.
