


On the
formal first cocycle equation for iteration groups of type
II 
On the formal first cocycle equation for iteration groups
of type II
Jointly written with LUDWIG REICH.
ESAIM: Proceedings 36, 3247, 2012. Proceedings of the
European Conference on Iteration Theory 2010.
Abstract: Let x be an indeterminate over ℂ. We
investigate solutions
α(s,x)=∑_{n≥ 0}
α_{n}(s)x^{n},^{.}_{.} 
α_{n}: ℂ→ ℂ, n≥ 0, of the first
cocycle equation
α(s+t,x)= α(s,x)α(t,F (s,x)),
s,t∈ ℂ, 
(Co1) 

in C[[x]], the ring of formal power series over ℂ,
where (F(s,x))_{s∈ ℂ} is an iteration group of
type II, i.e. it is a solution of the translation equation
F(s+t,x)=F(s,F(t,x)), s,t∈
ℂ, 
(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 α_{n}(s)
of
α(s,x)=1+∑_{n≥
1}α_{n}(s)x^{n}^{.}_{.} 
are polynomials in c_{k}(s).
It is possible to replace this additive function c_{k}
by an indeterminate. Finally, we obtain a formal version of the
first cocycle 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 of these polynomials depending on certain parameters, the
coefficients of the generator K of a formal cocycle for iteration
groups of type II. Rewriting the solutions Γ(y,x) of the
formal first cocycle equation in the form ∑_{n≥
1}ψ_{n}(x)y^{n} as elements of
(ℂ[[x]])[[y]], we obtain explicit formulas for
ψ_{n} in terms of the derivatives H^{(j)}(x)
and K^{(j)}(x) of the generators H and K and also a
representation of Γ(y,x) similar to a LieGröbner series.
There are interesting similarities between the solutions G(y,x) of
the formal translation equation for iteration groups of
type II and the solutions Γ(y,x) of the formal first
cocycle equation for iteration groups of type II.
harald.fripertinger "at" unigraz.at, October 3,
2024






GDPR 
On the
formal first cocycle equation for iteration groups of type
II 

