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

Summary

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:

$$F_t \circ G_t = G_t \circ F_t forallt \in \mathbb{C}.$$

((1))

Does (1) imply the stronger condition

$$F_t \circ G_s = G_s \circ F_t foralls,t \in \mathbb{C}?$$

((2))

(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. Leszczyńska.)

We give an affirmative answer to this question by characterizing all pairs (F_t)_t∈ℂ, (G_t)t∈ℂ of iteration groups which satisfy (1). For such pairs of iteration groups exactly two cases occur:

• (i) Both iteration groups can be simultaneously linearized (by conjugation).

• (ii) One of the iteration groups is a group with multiplier 1 and contains the other one as a subgroup.

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.