Anti-classification results for conjugacy of diffeomorphisms on manifolds

TL;DR

The paper establishes an anti-classification result for smooth dynamical systems: for any manifold with dimension at least 2, the topological conjugacy relation on $C^{\infty}$-diffeomorphisms is not classifiable by countable structures. It also proves that $E_0$ is reducible to the topological conjugacy relation of minimal diffeomorphisms on the $2$-torus, answering Foreman’s question and showing non-amenability of this conjugacy relation. The approach combines descriptive set-theoretic tools (Borel reductions, turbulence) with explicit diffeomorphism-construction techniques, notably two auxiliary function types $\psi$ and $m$ to realize controlled local rotations and block-moves. The results illuminate the limits of smooth classification and provide a concrete bridge from $E_0$ to smooth dynamical conjugacy on manifolds, leveraging a careful disk-pasting scheme and fixed-point analysis to separate conjugate and non-conjugate parameter families.

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.

Theorems & Definitions (21)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 2.1
• Lemma 3.1
• proof
• proof
• Theorem 5.1
• proof
• Lemma 5.2