On residues and conjugacies for germs of 1-D parabolic diffeomorphisms in finite regularity

TL;DR

This paper investigates the invariance of residues and conjugacy classes of 1-D parabolic diffeomorphisms with finite regularity, extending classical results to lower differentiability classes and analyzing the impact on conjugacy invariants.

Contribution

It establishes sharp results on conjugacy invariants for non-flat diffeomorphisms of the real line, extending the understanding of invariance under low-regular conjugacies.

Findings

Residues are invariant under low-regular conjugacies.

The Schwarzian derivative at the origin is invariant under $C^2$ conjugacy.

Invariants can vary arbitrarily under $C^1$ conjugacy.

Abstract

We study conjugacy classes of germs of non-flat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to the identity. The core of all of this lies in the invariance of residues under low-regular conjugacies. This may be seen as an extension of the fact (also proved in this article) that the value of the Schwarzian derivative at the origin for germs of $C^3$ parabolic diffeomorphisms is invariant under $C^2$ parabolic conjugacy, though it may vary arbitrarily under parabolic $C^1$ conjugacy.