  We investigate local entropy theory, particularly the property of having
completely positive entropy (CPE), from a descriptive set-theoretic point of
view. We aim to determine descriptive complexity of different families of
dynamical systems with CPE.
  For a large class of compact $X$, we show that the family of dynamical
systems on $X$ with CPE is complete coanalytic and hence not Borel. When we
restrict our attention to dynamical systems having special properties such as
the mixing property or the shadowing property, we obtain some contrasting
behavior. In particular, the notion of CPE and the notion of uniform positive
entropy, a Borel property, coincide for mixing maps on topological graphs. On
the other hand, the class of mixing map on the Cantor space is coanalytic and
not Borel.
  For dynamical systems with the shadowing property, the notions CPE and
uniform positive entropy coincide regardless of the phase space.
