## Nankai Logic Colloquium

Hello everyone,

This week our weekly Nankai Logic Colloquium is going to be in the afternoon.

Our speaker this week will be Dominik Kwietniak from Jagiellonian University. This talk is going to take place this Friday, June 2nd, from 4pm to 5pm(UTC+8, Beijing time).

Title: An anti-classification theorem for the topological conjugacy problem for Cantor minimal systems Abstract: 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 would like 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 Cantor set homeomorphism T is in Min(C) if every orbit of T is dense in C. We say that S and T in Min(C) are topologically conjugate if there exists a Cantor set homeomorphism h such that Sh=hT. 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 is 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) x Min(C) is complete analytic. In particular, TopConj is a non-Borel subset of Min(C) x Min(C). Roughly speaking, it means that 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 subtrees over non-negative integers N 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 with arbitrarily long branches is a well-known example of a complete analytic set, we see that it is essentially 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 Kapsrzak, Philipp Kunde (all from the Jagiellonian University in Kraków).

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Title ：The 31st Nankai Logic Colloquium --Dominik Kwietniak

Time ：16:00pm, Jun. 2, 2023(Beijing Time)

Zoom Number ： 876 3579 6414

Passcode ： 318535

Link ：https://us02web.zoom.us/j/87635796414?pwd=M1hZSEFvL0FzMUZQcHVCQ0w2QlhtUT09

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Best wishes,

Ming Xiao