Local entropy theory and descriptive complexity

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.