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arXiv · 2209.02137

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

Philipp Kunde
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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.

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Philipp Kunde. 2022-09-05. There is no complete numerical invariant for smooth conjugacy of circle diffeomorphisms. https://arxiv.org/abs/2209.02137

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