There is no complete numerical invariant for smooth conjugacy of circle diffeomorphisms

TL;DR

This paper demonstrates that no complete numerical invariant exists for classifying smooth conjugacy of circle diffeomorphisms beyond topological conjugacy, using advanced techniques from Descriptive Set Theory and approximation methods.

Contribution

It proves the non-existence of a complete numerical Borel invariant for smooth conjugacy of circle diffeomorphisms with higher regularity.

Findings

No numerical invariant classifies smooth conjugacy of circle diffeomorphisms.

The proof uses Descriptive Set Theory and Approximation by Conjugation techniques.

Results extend to H"older and $C^k$ conjugacies.

Abstract

Classical results by Poincar\'e and Denjoy show that two orientation-preserving $C^2$ diffeomorphisms of the circle are topologically conjugate if and only if they have the same rotation number. We show that there is no possibility of getting such a complete numerical Borel invariant for the conjugacy relation of orientation-preserving circle diffeomorphisms by homeomorphisms with higher degree of regularity. For instance, we consider conjugacy by H\"older homeomorphisms or by $C^k$-diffeomorphisms with $k\in \mathbb{Z}^+ \cup \{\infty\}$. The proof combines techniques from Descriptive Set Theory and a quantitative version of the Approximation by Conjugation method for circle diffeomorphisms.