Commentarii Mathematici Helvetici

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Volume 86, Issue 2, 2011, pp. 415–435
DOI: 10.4171/CMH/229

Published online: 2011-02-27

On the centralizer of diffeomorphisms of the half-line

Hélène Eynard[1]

(1) Université de Lyon, Lyon, France

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.

Keywords: Interval diffeomorphism, centralizer, commuting, Liouville number, vector field, flow

Eynard Hélène: On the centralizer of diffeomorphisms of the half-line. Comment. Math. Helv. 86 (2011), 415-435. doi: 10.4171/CMH/229