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

Hélène Eynard-Bontemps, Andrés Navas

TL;DR

This work analyzes germs of real 1-D parabolic diffeomorphisms fixing the origin through algebraic residues and their dynamical incarnations. It develops a precise algebraic framework with groups of formal diffeomorphisms, introduces additive residues $\\mathrm{Resad}_{\ell}$ and the iterative residue $\\mathrm{Resit}$, and links these to a higher-order Schwarzian derivative, providing a hierarchy of invariants under low-regularity conjugacies. The authors prove that $\\mathrm{Resit}$ is invariant under $C^{\ell+1}$ conjugacies and that this threshold is sharp, while also giving sharp existence results for low-regular conjugacies in finite regularity and detailing how residues influence or obstruct conjugacies. They illustrate the sharpness with explicit examples, connect residues to associated vector fields and their flows, and discuss implications for distortion in groups of germs, highlighting rigidity phenomena and revealing both general and delicate interactions between regularity, residues, and dynamics. Overall, the paper clarifies how residues govern rigidity in 1-D parabolic germ conjugacies and extends Takens–Yoccoz-type results to finite differentiability, with clear implications for the structure of germ groups.

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.

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

TL;DR

This work analyzes germs of real 1-D parabolic diffeomorphisms fixing the origin through algebraic residues and their dynamical incarnations. It develops a precise algebraic framework with groups of formal diffeomorphisms, introduces additive residues and the iterative residue , and links these to a higher-order Schwarzian derivative, providing a hierarchy of invariants under low-regularity conjugacies. The authors prove that is invariant under conjugacies and that this threshold is sharp, while also giving sharp existence results for low-regular conjugacies in finite regularity and detailing how residues influence or obstruct conjugacies. They illustrate the sharpness with explicit examples, connect residues to associated vector fields and their flows, and discuss implications for distortion in groups of germs, highlighting rigidity phenomena and revealing both general and delicate interactions between regularity, residues, and dynamics. Overall, the paper clarifies how residues govern rigidity in 1-D parabolic germ conjugacies and extends Takens–Yoccoz-type results to finite differentiability, with clear implications for the structure of germ groups.

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 parabolic diffeomorphisms is invariant under parabolic conjugacy, though it may vary arbitrarily under parabolic conjugacy.
Paper Structure (16 sections, 19 theorems, 176 equations)

This paper contains 16 sections, 19 theorems, 176 equations.

Table of Contents

  1. Some general (algebraic) facts
  2. Germs and morphisms
  3. Res-it-ad
  4. Schwarzian derivatives and $\mathrm{Resad}_{\ell}$
  5. On conjugacies and residues: statements, examples
  6. A fundamental example: (non-)invariance of residues
  7. Examples of low regular conjugacies that preserve residues
  8. $C^{2\ell+1}$ germs with vanishing residue that are non $C^{\ell+1}$ conjugate
  9. On the conjugacy invariance of residues
  10. Residues and vector fields
  11. A first proof of the $C^{\ell + 1}$ conjugacy invariance of $\mathrm{Resit}$
  12. Residues and logarithmic deviations of orbits
  13. A second proof of the $C^{\ell + 1}$ conjugacy invariance of $\mathrm{Resit}$
  14. Existence of low regular conjugacies between germs
  15. Conjugacies in case of coincidence of residues
  16. ...and 1 more sections

Key Result

Theorem A

Given $\ell \geq 1$, let $f,g$ be two parabolic germs in $\mathrm{Diff}^{2\ell + 1}_+(\mathbb{R},0)$ that are exactly $\ell$-tangent to the identity. If $f$ and $g$ are conjugated by a germ in $\mathrm{Diff}^{\ell + 1}_+ (\mathbb{R},0)$, then they have the same (iterative) residue.

Theorems & Definitions (51)

  • Theorem A
  • Theorem B
  • Theorem B'
  • Theorem C
  • Lemma 1.1
  • Example 1.2
  • Lemma 1.3
  • proof
  • Remark 1.4
  • Remark 1.5
  • ...and 41 more