Remarks on iteration of formal automorphisms (Q1112881)

If R is a homomorphic image of a formal power series ring \({\mathbb{C}}[[z_ 1,...,z_ n]]\) over the field \({\mathbb{C}}\) of complex numbers, then R is a local ring with maximal ideal m generated by the residue classes of \(z_ 1,...,z_ n\). For each nonnegative integer k, \(R/m^{k+1}\) is a finite dimensional \({\mathbb{C}}\)-algebra, there is a natural homomorphism \(\pi_ k\) of the group A of automorphisms of the \({\mathbb{C}}\)-algebra R into the group \(A_ k\) of automorphisms of the \({\mathbb{C}}\)-algebra \(R/m^{k+1}\), and \(A_ k\) is an algebraic matrix group. An element \(\phi\) of A is called analytically iterable if there is a homomorphism \(\eta\) of the additive group of \({\mathbb{C}}\) into A such tha all \(\pi_ k\circ \eta:\quad {\mathbb{C}}\to A_ k\) are analytic and \(\eta (1)=\phi\). If D is the Lie algebra of \({\mathbb{C}}\)-linear derivations of R mapping m into itself, there is a map \(\exp:\quad D\to A\) which induces the exponential map of the Lie algebra of \({\mathbb{C}}\)-linear derivations of \(R/m^{k+1}\) into \(A_ k\). The author shows that an element \(\phi\) of A is analytically iterable if and only if \(\phi\in \exp (D)\). Also the author uses the theory of algebraic groups to show that automorphisms of several types are analytically iterable.