Anti-classification results for conjugacy of diffeomorphisms on manifolds

TL;DR

This paper demonstrates that classifying diffeomorphisms on manifolds of dimension two or higher by topological conjugacy is highly complex, not reducible to countable structures, and relates this complexity to the reducibility of a known equivalence relation.

Contribution

It establishes the non-classifiability of topological conjugacy for diffeomorphisms on higher-dimensional manifolds and connects this to the reducibility of $E_0$ into minimal diffeomorphisms on the 2-torus.

Findings

Topological conjugacy of diffeomorphisms on manifolds of dimension ≥ 2 is not classifiable by countable structures.

The equivalence relation $E_0$ reduces to the conjugacy relation of minimal diffeomorphisms on the 2-torus.

Answers open questions posed by Foreman and Gorodetski.

Abstract

We show that the topological conjugacy relation of diffeomorphisms on any manifold of dimension at least 2 is not classifiable by countable structures. This answers a question of Foreman and Gorodetski. We also prove that $E_0$ is reducible into the topological conjugacy relation of minimal diffeomorphisms on the 2-torus, which answers a question of Foreman.