A fundamental theme in dynamics is the classification of systems up to appropriate equivalence relations. For instance, the equivalence relation of topological conjugacy preserves the qualitative behavior of topological dynamical systems. Smale’s celebrated program proposes to classify topological or smooth dynamical systems up to topological conjugacy.

These classification problems not only turn out to be hard but sometimes even to be impossible. In joint work with Deka, Garcia-Ramos, Kasprzak, and Kwietniak, we show that the equivalence relation generated by topological conjugacy of minimal homeomorphisms on a Cantor space is not a Borel set. This implies that Cantor minimal systems cannot be classified using inherently countable techniques.