Über ein Problem von J. Schwaiger und Z. Moszner betreffend die Vertauschbarkeit von einparametrigen Automorphismengruppen formaler Potenzreihenringe

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Let ℂ〚x〛 be the ring of formal power series in one indeterminate over ℂ, and let Γ be the group of automorphisms of ℂ〚x〛 which are continuous in the order topology and leave ℂ elementwise fixed. Assume that(F t ) t ∈ ℂ and(G t ) t ∈ ℂ are iteration groups, i.e. one-parameter subgroups of Γ which are solutions of the translation equationF t ∘F s =F t + s ,G t ∘G s =G t + s . Suppose moreover that the following weak commutativity condition holds: (1) $$F_t \circ G_t = G_t \circ F_t forallt \in \mathbb{C}.$$ Does (1) imply the stronger condition (2) $$F_t \circ G_s = G_s \circ F_t foralls,t \in \mathbb{C}?$$ (This problem had been posed by J. Schwaiger. Similar problems for homomorphisms of (ℂ, + ) into groups of matrices have been dealt with by Z. Moszner and Z. Leszczynska.) We give an affirmative answer to this question by characterizing all pairs (Ft)t∈ℂ, (Gt)t∈ℂ of iteration groups which satisfy (1). For such pairs of iteration groups exactly two cases occur: We do not assume that the iteration groups under consideration are analytic. Indeed, no assumption on the regularity of the dependence on the group parameter is made.