On the centralizer of diffeomorphisms of the half-line

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 infinite cyclic 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.