DSNT Seminar: Classification of mixing SFTs without periodic points

Abstract: By Ornstein's theorem (and its later extensions), entropy gives a full classification for mixing Markov shifts (in the category of measure-theoretic dynamical systems). It is easy to show that the topological analog does not directly hold, in that mixing SFTs are not fully classified by entropy: the binary full shift has the same entropy as the set of graph homomorphisms from the integers to a triangle, but these systems are not topologically conjugate (because their periodic points do not agree). However, entropy is a reasonably good invariant in the topological category as well: If one allows removing null sets, entropy does classify mixing SFTs up to topological conjugacy. It is also known that if one removes just the periodic points, entropy classifies mixing SFTs up to Borel isomorphism. It was asked by Hochmann whether there is a common generalization, i.e. whether two mixing SFTs with the same entropy become topologically conjugate as soon as we remove their periodic points. In the talk, we explain this history in some more detail, and present a complete* solution to Hochmann's question.

This is a seminar in our seminar series on Dynamical Systems and Number Theory (DSNT).