  We show that for a non-trivial transitive dynamical system, it has a dense
Mycielski invariant strongly scrambled set if and only if it has a fixed point,
and it has a dense Mycielski invariant $\delta$-scrambled set for some
$\delta>0$ if and only if it has a fixed point and not uniformly rigid. We also
provide two methods for the construction of completely scrambled systems which
are weakly mixing, proximal and uniformly rigid.
