Centralisers of formal maps

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 $\mathcal{G}$. We consider the centraliser $C_g$ of an element $g\in\mathcal{G}$ which is tangent to the identity of $\mathcal{G}$. Elements of finite order always have an uncountable centraliser. If $g$ has infinite order and $K$ is a field of characteristic zero we show that $C_g$ contains an isomorphic copy of the additive group $(K,+)$. If $g$ has infinite order and $K$ has finite characteristic we show that $C_g$ contains an uncountable abelian subgroup. The proofs are quite different in finite characteristic and in characteristic zero, but are connected by so-called sum functions.

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