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On Problem 32 from Rufus Bowen's list: classification of shift spaces with specification

Banff International Research Station for Mathematical Innovation and Discovery

Rufus Bowen left a notebook containing 157 open problems and questions. Problem 32 on that list asks for classification of shift spaces with the specification property. Unfortunately, there is no universally accepted agreement what does it mean âto classifyâ a family of mathematical objects, and Bowen didn't left any clues. During my talk, I will describe one of the most popular ways of making the problem formal. It is based on the theory of Borel equivalence relations. Inside that framework, I will explain a result saying that (roughly speaking) there is no reasonable classification for shift spaces with the specification property. More precisely, I will show that the isomorphism relation on the space of shifts with the specification property is a universal countable Borel equivalence relation, i.e. for every countable Borel equivalence relation $F$, we have that $F$ is Borel reducible to $E$. It follows that no classification using a finite set of definable invariants is possible. This solves the problem provided that Bowen would agree with the notion of âclassificationâ provided by the theory of Borel equivalence relations.

Mathematics; Dynamical systems and ergodic theory; Group theory and generalizations; Dynamical systems

video/mp4

Author affiliation: Jagiellonian University

2019-11-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72248

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/