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All, most, some? On diffeomorphisms of the interval that are distorted and/or conjugate to powers of themselves

Part of: Low-dimensional dynamical systems Smooth dynamical systems: general theory Low-dimensional topology

Published online by Cambridge University Press:  02 April 2025

Hélène Eynard-Bontemps Open the ORCID record for Hélène Eynard-Bontemps [Opens in a new window]  and
Andrés Navas Open the ORCID record for Andrés Navas [Opens in a new window]
Hélène Eynard-Bontemps*
Affiliation:
Institut Fourier, Laboratoire de Mathématiques, Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France,
Andrés Navas
Affiliation:
Dpto. de Matemáticas y C.C., Universidad de Santiago de Chile, Alameda Bernardo O’Higgins, Estación Central, Santiago, Chile
*
Corresponding author: Hélène Eynard-Bontemps, email: helene.eynard-bontemps@univ-grenoble-alpes.fr
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Abstract

We study the problem of conjugating a diffeomorphism of the interval to (positive) powers of itself. Although this is always possible for homeomorphisms, the smooth setting is rather interesting. Besides the obvious obstruction given by hyperbolic fixed points, several other aspects need to be considered. As concrete results we show that, in class C1, if we restrict to the (closed) subset of diffeomorphisms having only parabolic fixed points, the set of diffeomorphisms that are conjugate to their powers is dense, but its complement is generic. In higher regularity, however, the complementary set contains an open and dense set. The text is complemented with several remarks and results concerning distortion elements of the group of diffeomorphisms of the interval in several regularities.

Keywords

diffeomorphismconjugacyMather invariantasymptotic variationdistortion element

MSC classification

Primary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
Secondary: 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 57M60: Group actions in low dimensions

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 36
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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