


On
covariant embeddings of a linear functional equation with respect
to an analytic iteration group 
On covariant embeddings of a linear functional equation
with respect to an analytic iteration group
Jointly written with LUDWIG REICH.
International
Journal of Bifurcation and Chaos (World Scientific Publishing
Co.) Vol. 13, No. 7, 1853  1875, 2003.
Abstract: Let a(x), b(x), p(x) be formal power series in
the indeterminate x over ℂ (i.e. elements of the ring
C[[x]] of such series), such that orda(x)=0, ordp(x)=1 and
p(x) is embeddable into an analytic iteration group
(π(s,x))_{s ∈ ℂ} in C[[x]]. By a
covariant embedding of the linear functional equation
φ(p(x))=a(x)φ(x)+b (x), 
(L) 

(for the unknown series φ(x) inC[[x]] ) with respect
to (π(s,x))_{s ∈ ℂ} we understand families
(α(s,x))_{s ∈ ℂ} and (β(s,x))_{s
∈ ℂ}
with entire coefficient functions in s, such that the system of
functional equations and boundary conditions
φ(π(s,x))=α(s,x)φ(x)+β(s,x) 
(Ls) 

α(t+s,x)= α(s,x)α(t,π(s,x)) 
(Co1) 

β(t+s,x)=
β(s,x)α(t,π(s,x)) +β(t,π(s,x)) 
(Co2) 

α(1,x)=a(x) β(1,x)=b(x) 
(B2) 

holds for all solutions φ(x) of (L) and s,t in ℂ. In
this paper we solve the system ( (Co1),(Co2)) (of so called cocycle
equations) completely, describe when and how the boundary
conditions (B1) and (B2) can be satisfied and present a large class
of equations (L) together with iteration groups (π(s,x))_{s
∈ ℂ} for which there exist covariant embeddings of
(L) with respect to (π(s,x))_{s ∈ ℂ}.
harald.fripertinger "at" unigraz.at, October 12,
2017






On
covariant embeddings of a linear functional equation with respect
to an analytic iteration group 

