Über die multiplikative Translationsgleichung und idempotente Potenzreihenvektoren

Abstract

We consider the functional equation

$$\Phi (x,\Phi (y,z)) = \Phi (x'y,z)$$

as the identity for vectors of power series φ(x, z):=φ(x; z₁, ..., z _n ) in the variables x, z₁, ..., z _in with coefficients from a commutative ring with unit element and an additional condition, which is satisfied, e.g. for principle ideal domains (fields, polynomial rings or rings of power series in one variable over a field). The above equation is of interest in the iteration theory. All the solutions of this equation can be determined. As a corollary, one gets a complete description of the idempotent vectors of power series φ(z₁, ..., z _n ), i.e. the vectors of power series which satisfy φ.φ=φ.