


The formal
translation equation for iteration groups of type II 
The formal translation equation for iteration groups of
type II
Jointly written with LUDWIG REICH.
Aequationes
mathematicae 79, 111156, 2010.
Abstract: We investigate the translation equation
F(s+t,x)=F(s,F(t,x)), s,t∈
ℂ, 
(T) 

in C[[x]], the ring of formal power series over ℂ.
Here we restrict ourselves to iteration groups of type II,
i.e. to solutions of (T) of the form F(s,x)≡
x+c_{k}(s)x^{k}modx^{k+1}, where k≥ 2
and c_{k} ≠ 0 is necessarily an additive function. It is
easy to prove that the coefficient functions c_{n}(s) of
F(s,x)=x+^{.}_{.} 
∑ n≥ k 
c_{n}(s)x^{n}^{.}_{.} 
are polynomials in c_{k}(s).
It is possible to replace this additive function c_{k}
by an indeterminate. In this way we obtain a formal version of the
translation equation in the ring (ℂ[y])[[x]]. We solve this
equation in a completely algebraic way, by deriving formal
differential equations or an AczélJabotinsky type equation. This
way it is possible to get the structure of the coefficients in
great detail which are now polynomials. We prove the universal
character (depending on certain parameters, the coefficients of the
infinitesimal generator H of an iteration group of type II) of
these polynomials. Rewriting the solutions G(y,x) of the formal
translation equation in the form ∑_{n≥
0}φ_{n}(x)y^{n} as elements of
(ℂ[[x]])[[y]], we obtain explicit formulas for
φ_{n} in terms of the derivatives H^{(j)}(x) of
the generator H and also a representation of G(y,x) as a
LieGröbner series. Eventually, we deduce the canonical form (with
respect to conjugation) of the infinitesimal generator H as
x^{k}+hx^{2k1} and find expansions of the
solutions G(y,x)=∑_{r≥ 0}
G_{r}(y,x)h^{r} of the above mentioned differential
equations in powers of the parameter h.
harald.fripertinger "at" unigraz.at, March 4,
2019






GDPR 
The formal
translation equation for iteration groups of type II 

