Centralisers of Formal Maps

Abstract

We consider the formal maps in any finite dimension d with coefficients in an integral domain K with identity. Those invertible under formal composition form a group [inline-graphic 1i]. We consider the centraliser Cg of an element g ∈ [inline-graphic 1i] of infinite order, tangent to the identity of [inline-graphic 1i]. If g has infinite order and K is a field of characteristic zero, we show that Cg contains an isomorphic copy of the additive group (K, +). If g has infinite order and K has positive characteristic, we show that Cg 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.