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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" uni-graz.at, October 3,
2024
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GDPR |
On
covariant embeddings of a linear functional equation with respect
to an analytic iteration group |
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