Title: The characteristic measure problem for subshifts
Abstract: the Krylov-Bogolubov theorem guarantees that every topological Z-system has a measure invariant under the dynamics. However, in many systems, there are additional symmetries that form a nontrivial group called its automorphism group. In 2019, Frisch and Tamuz introduced the characteristic measure problem for subshufts: does every symbolic Z-system have a measure invariant invariant under its full automorphism group? Interestingly, the answer to the characteristic measure problem is negative for non-symbolic systems, but there is a growing literature showing that an extremely broad class of symbolic systems have characteristic measures. In this talk, I will survey some recent joint work with Solly Coles and Bryna Kra in which we make progress on this problem for subshifts.