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

2023/02/25 by Hélène Eynard-Bontemps, Eynard-Bontemps, Hélène, Andrés Navas +1 · 1 citation
Mathematics · #20F38 #37C25 #37E99 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2302.13182

openalex publication_date 2023/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 C³ parabolic diffeomorphisms is invariant under C² parabolic conjugacy, though it may vary arbitrarily under parabolic C¹ conjugacy.

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