  There are lots of results to study dynamical complexity on irregular sets and
level sets of ergodic average from the perspective of density in base space,
Hausdorff dimension, Lebesgue positive measure, positive or full topological
entropy (and topological pressure) etc.. However, it is unknown from the
viewpoint of chaos. There are lots of results on the relationship of positive
topological entropy and various chaos but it is known that positive topological
entropy does not imply a strong version of chaos called DC1 so that it is
non-trivial to study DC1 on irregular sets and level sets. In this paper we
will show that for dynamical system with specification property, there exist
uncountable DC1-scrambled subsets in irregular sets and level sets. On the
other hand, we also prove that several recurrent levels of points with
different recurrent frequency all have uncountable DC1-scrambled subsets. The
main technique established to prove above results is that there exists
uncountable DC1-scrambled subset in saturated sets.
