On the Distribution of Formal Power Series Transformationswith Respect to Embeddability in the OrderTopology

Let C[[x1, . . . ,xn]] (brie£y C [[x]], where xˆ (x1, . . . ,xn) is the vector of indeterminates) be the ring of formal power series in n indeterminates x1, . . . ,xn with complex coe¤cients.We consider in this paper formal power series transformations F by which we understand automorphisms F of C[[x]] which are continuous with respect to the order topology (i.e., order preserving) and leave every element of the ground ¢eld C ¢xed. It is well known that these automorphisms F are in 1ÿ1 correspondence to the images F…x† ˆ Ax‡ P…x† of x. Here A runs through the matrices of GL (n,C), and P…x† is an n-tuple of formal power series with ord…P† 2. Moreover, these automorphisms form a group ÿ under composition which is, in the above mentioned picture, represented by substitution of one n-tupel Ax‡P…x† 2 C‰‰xŠŠ into another. F is called iterable (embeddable), if there exists a family (Ft)t2C in ÿ such that