Work

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

Year: 2025

Type: article

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.

Source: Proceedings of the Edinburgh Mathematical Society

DOI: 10.1017/s0013091525000148

Authors: Hélène Eynard-Bontemps, Andrés Navas

Institutions: Institut Fourier, Université Grenoble Alpes, Universidad de Santiago de Chile, University of Chile

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Agence Nationale de la Recherche, Centre National de la Recherche Scientifique, Institut Henri Poincaré, Sorbonne Université