On Problem 32 from Rufus Bowen’s list: classification of shift spaces with specification
Rufus Bowen left a notebook containing 157 open problems and questions. Problem 32 on that list asks for a classification of shift spaces with the specification property. Unfortunately, there is no agreement what does it mean “to classify” a family of mathematical objects. I will describe one of the most commonly accepted ways of making the problem formal based on the language of Borel equivalence relations. Inside that framework I will explain a theorem saying that (roughly speaking) there is no reasonable classification for shift spaces with specification. In particular, no classification using a finite set of definable invariants is possible. This solves the problem provided that Bowen would agree with making the notion of “classification” rigorous through set theory.