


The formal
translation equation and formal cocycle equations for iteration
groups of type I 
The formal translation equation and formal cocycle
equations for iteration groups of type I
Jointly written with LUDWIG REICH.
Aequationes
mathematicae 76, 54  91, 2008.
Abstract: We investigate the translation equation
F(s+t,x)=F(s,F(t,x)), s,t∈
ℂ, 
(T) 

and the cocycle equations
α(s+t,x)= α(s,x)α(t,F (s,x)),
s,t∈ ℂ, 
(Co1) 

β(s+t,x)=
β(s,x)α(t,F (s,x)) +β(t,F (s,x)),
s,t∈ ℂ 
(Co2) 

in C[[x]], the ring of formal power series over ℂ.
Here we restrict ourselves to iteration groups (F(s,x))_{s∈
ℂ} of type I, i.e. to solutions of (T) of the form
F(s,x)=c_{1}(s)x+…, where c_{1} ≠ 1 is a
generalized exponential function. It is easy to prove that the
coefficient functions c_{n}(s), α_{n}(s),
β_{n}(s) of
F(s,x)=^{.}_{.} 
∑ n≥ 1 
c_{n}(s)x^{n},
α(s,x)=^{.}_{.} 
∑ n≥ 0 
α_{n}(s)x^{n},
β(s,x)=^{.}_{.} 
∑ n≥ 0 
β_{n}(s)x^{n}^{.}_{.} 
are polynomials in c_{1}(s), polynomials or rational
functions in one more generalized exponential function and, in a
special case, polynomials in an additive function. For example, we
obtain c_{n}(s)=P_{n}(c_{1}(s)),
P_{n}(y)∈ ℂ[y], n≥ 1. For this we do not need
any detailed information about the polynomials P_{n}.
Under some conditions on the exponential and additive functions
it is possible to replace the exponential and additive functions by
independent indeterminates. In this way we obtain formal versions
of the translation equation and the cocycle equations in rings of
the form (ℂ[y])[[x]], (ℂ[S,σ])[[x]],
(ℂ(S))[[x]], (ℂ(S)[U])[[x]], and
(ℂ(S)[σ,U])[[x]]. We solve these equations in a
completely algebraic way, by deriving formal differential equations
or AczélJabotinsky type equations for these problems. It is
possible to get the structure of the coefficients in great detail
which are now polynomials or rational functions. We prove the
universal character (depending on certain parameters) of these
polynomials or rational functions. And we deduce the canonical form
S^{1}(yS(x)) for iteration groups of type I. This
approach seems to be more clear and more general than the original
one. Some simple substitutions allow us to solve these problems in
rings of the form (ℂ[[u]])[[x]], i. e. where the
coefficient functions are formal series.
harald.fripertinger "at" unigraz.at, October 12,
2017






The formal
translation equation and formal cocycle equations for iteration
groups of type I 

