Dynamical Systems Seminar

Slow entropy is a family of invariants used to study zero-entropy systems, both topological and measure-preserving. A variational principle sometimes holds, and sometimes it fails. The talk will be about systems where this variational principle fails.

By a theorem of Clemens, topological conjugacy of subshifts is a universal countable Borel equivalence relation. We bound from above and below the Borel complexity of its restrictions to minimal subshifts and to subshifts with the specification property. For minimal subshifts, conjugacy is known to be nonsmooth, and Gao asks whether it is hyperfinite or universal. For subshifts with specification, joint work with Konrad Deka, Bo Peng and Marcin Sabok shows that conjugacy is not treeable. Hence Bowen's Problem 32, which asks for a classification of these subshifts, admits no solution by concrete invariants. We also present a proof that neither relation is universal. Thus conjugacy of subshifts with specification is neither treeable nor universal, and conjugacy of minimal subshifts is nonsmooth and not universal. Whether it is hyperfinite remains open.