TY - ELEC
AU - Dominik Kwietniak
PY - 2019
TI - On Problem 32 from Rufus Bowen's list: classification of shift spaces with specification
LA - eng
M3 - Moving Image
AB - 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 Ã¢to classifyÃ¢ 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
Ã¢classificationÃ¢ provided by the theory of Borel equivalence
relations.
N2 - 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 Ã¢to classifyÃ¢ 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
Ã¢classificationÃ¢ provided by the theory of Borel equivalence
relations.
UR - https://open.library.ubc.ca/collections/48630/items/1.0385160
ER - End of Reference