Anti-classification results for smooth dynamical systems. Preliminary Report

TL;DR

This paper investigates the complexity of classifying smooth dynamical systems up to conjugacy, showing that in higher dimensions, such classification is highly non-trivial and cannot be achieved through simple invariants.

Contribution

It demonstrates the non-Borel and complete analytic nature of conjugacy classification for diffeomorphisms in dimensions 2 and above, advancing understanding of dynamical systems complexity.

Findings

No Borel method exists for complete invariants in dimension 2+

Conjugacy relation is not Borel in dimension 5+

The classification problem is complete analytic in high dimensions

Abstract

The paper considers the equivalence relation of conjugacy-by-homeomorphism on diffeomorphisms of smooth manifolds. In dimension 2 and above it is shown that there is no Borel method of attaching complete numerical invariants. In dimension 5 and above it is shown that the equivalence relation is not Borel, and in fact is complete analytic.