Talk
An anti-classification theorem for the topological conjugacy of Cantor minimal systems
Talk by Dr. Dominik Kwietniak
Date: 30.10.23 Time: 15.00 - 16.00 Room: Y27H25
The isomorphism problem in dynamics dates back to a question of von Neumann from 1932: Is it possible to classify (in some reasonable sense) the ergodic measure-preserving diffeomorphisms of a compact manifold up to isomorphism? We want to study a similar problem: Let C be the Cantor set and let Min(C) stand for the space of all minimal homeomorphisms of the Cantor set. Recall that a homeomorphism f is in Min(C) if every orbit of f is dense in C. We say that f and g in Min(C) are topologically conjugate if there is a Cantor set homeomorphism h such that f o h = h o g. We prove an anti-classification result showing that even for very liberal interpretations of what a "reasonable'' classification scheme might be, a classification of minimal Cantor set homeomorphism up to topological conjugacy is impossible. We see it as a consequence of the following: we prove that the topological conjugacy relation of Cantor minimal systems TopConj treated as a subset of Min(C)xMin(C) is complete analytic. In particular, TopConj is a non-Borel subset of Min(C)xMin(C). Roughly speaking, it is impossible to tell if two minimal Cantor set homeomorphisms are topologically conjugate using only a countable amount of information and computation.
Our result is proved by applying a Foreman-Rudolph-Weiss-type construction used for an anti-classification theorem for ergodic automorphisms of the Lebesgue space. We find a continuous map F from the space of all trees over non-negative integers with arbitrarily long branches into the class of minimal homeomorphisms of the Cantor set. Furthermore, F is a reduction, which means that a tree T is ill-founded if and only if F(T) is topologically conjugate to its inverse. Since the set of ill-founded trees is a well-known example of a complete analytic set, it is impossible to classify which minimal Cantor set homeomorphisms are topologically conjugate to their inverses.
This is joint work with Konrad Deka, Felipe García-Ramos, Kosma Kasprzak, Philipp Kunde (all from the Jagiellonian University in Kraków).