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ON RESIDUES AND CONJUGACIES FOR GERMS OF 1-D PARABOLIC DIFFEOMORPHISMS IN FINITE REGULARITY

Part of: Special aspects of infinite or finite groups Smooth dynamical systems: general theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  01 December 2023

Hélène Eynard-Bontemps  and
Andrés Navas Open the ORCID record for Andrés Navas [Opens in a new window]
Hélène Eynard-Bontemps
Affiliation:
Université Grenoble Alpes, CNRS, Institut Fourier, 38000 Grenoble, France and Center for Mathematical Modeling, FCFM, Universidad de Chile, Santiago, Chile (helene.eynard-bontemps@univ-grenoble-alpes.fr)
Andrés Navas*
Affiliation:
Dpto. de Matemática y C.C., Universidad de Santiago de Chile, Alameda Bernardo O’Higgins 3363, Estación Central, Santiago, Chile
*
andres.navas@usach.cl
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Abstract

We study conjugacy classes of germs of nonflat 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.

MSC classification

Primary: 20F38: Other groups related to topology or analysis 37C25: Fixed points, periodic points, fixed-point index theory 37E99: None of the above, but in this section

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 4 , July 2024 , pp. 1821 - 1855
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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