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

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(0,x)=1         β(0,x)=0	(B1)

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}.

• tth
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GDPR

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