Über ein Problem von J. Schwaiger und Z. Moszner betreffend die Vertauschbarkeit von einparametrigen Automorphismengruppen formaler Potenzreihenringe. (On a problem of J. Schwaiger and Z. Moszner concerning the commutativity of one-parameter groups of automorphisms of rings of formal power series) (Q593614)

The following question has some interest: Let \(\phi\) and \(\psi\) be two homomorphisms of groups defined on \({\mathbb{R}}\) or \({\mathbb{C}}\) with values in some group G. Suppose that \(\phi(t)\) commutes with \(\psi(t)\) for all t. Is it then true, that \(\phi(t)\) commutes with \(\psi\) (s) for all s and t? In this paper the case is considered that G is the group Aut\(({\mathbb{C}}[[X]])\) of order preserving automorphisms of the ring of formal power series in one indeterminate. It is shown that the answer to the question above is affirmative; moreover all possible cases are characterized in the following. Theorem: Let \((F_ t)_{t\in {\mathbb{C}}}\) and \((G_ t)_{t\in {\mathbb{C}}}\) be two one-parameter subgroups of Aut\(({\mathbb{C}}[[X]])\). Then \(F_ t\circ G_ t=G_ t\circ F_ t\) for all t if and only if either there is some T such that \(T^{-1}\circ F_ t\circ T\) and \(T^{-1}\circ G_ t\circ T\) are of the form u(t)X and v(t)X resp. or both subgroups have linear part 1 (i.e. \(F_ t=X+...,G_ t=X+...)\) and one of them, say \((F_ t)_{t\in {\mathbb{C}}}\), is non trivial and furthermore there is some \(t_ 0\) such that \((G_ t)_{t\in {\mathbb{C}}}\) is a subgroup of the analytic iteration group generated by \(F_{t_ 0}\neq id\). In both cases \(F_ t\) and \(G_ s\) commute for all s and t. It is worthwhile to note, that there are examples where the answer to the question is negative; especially there are examples, which imply that the answer is negative for Aut\(({\mathbb{C}}[[ X_ 1,X_ 2,...,X_ n]]\) with \(n\geq 4\). The cases \(n=2,3\) are unsolved.