On the centralizer of diffeomorphisms of the half-line

TL;DR

This paper investigates the structure of centralizers of smooth diffeomorphisms of the half-line, revealing that they can be dense, uncountable, and more complex than previously understood, especially for higher regularity classes.

Contribution

It demonstrates that the centralizer of a diffeomorphism can be a dense, uncountable subgroup of the known one-parameter group, expanding the understanding of their possible structures.

Findings

Centralizers can be dense and uncountable in the group of diffeomorphisms.

The phenomenon of complex centralizers is more common than previously thought.

Examples exist where the centralizer reduces to the group generated by the diffeomorphism.

Abstract

Let f be a smooth diffeomorphism of the half-line fixing only the origin and Z^r its centralizer in the group of C^r diffeomorphisms. According to well-known results of Szekeres and Kopell, Z^1 is a one-parameter group. On the other hand, Sergeraert constructed an f whose centralizer Z^r, $2\le r\le \infty$, reduces to the group generated by f. We show that Z^r can actually be a proper dense and uncountable subgroup of Z^1 and that this phenomenon is not scarce.