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

Classical results by Poincaré 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 conjuga...

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Bibliographic Details
Published in:arXiv.org 2022-09
Main Author: Kunde, Philipp
Format: Article
Language:English
Subjects:
Conjugation
Invariants
Isomorphism
Set theory
Topology
Online Access:Get full text
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Summary:Classical results by Poincaré 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.
ISSN:2331-8422