Centralisers of Formal Maps

Abstract

We consider formal maps in any finite dimension $d$ with coefficients in an integral domain $K$ with identity. Those invertible under formal composition form a group $\G$. We consider the centraliser $C_g$ of an element $g\in\G$ which is tangent to the identity of $\G$. Elements of finite order always have a large centraliser. If $g$ has infinite order our main result is that $C_g$ is uncountable, and in fact contains an uncountable abelian subgroup. This holds regardless of the characteristic of $K$, but the proof is quite different in finite characteristic than in characteritic zero.