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

TL;DR

This paper investigates when interval diffeomorphisms can be conjugated to their powers, revealing that in $C^1$ regularity such conjugations are dense but not generic, while in higher regularity the opposite holds.

Contribution

It characterizes the density and genericity of diffeomorphisms conjugate to their powers across different regularity classes, highlighting the role of fixed points and distortion.

Findings

In $C^1$, conjugate-to-power diffeomorphisms are dense among those with only parabolic fixed points.

In higher regularity, the set of diffeomorphisms not conjugate to their powers is open and dense.

The study includes remarks on distortion elements in the group of interval diffeomorphisms.

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 $C^1$, if we restrict to the (closed) subset of diffeomorphisms having only parabolic fixed points, then 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.