Tiling problems in music theory. Publications, to be read Remarks on Rhythmical Canons On the general solution of the system of cocycle equations without regularity conditions

On the general solution of the system of cocycle equations without regularity conditions

Jointly written with LUDWIG REICH.

Aequationes mathematicae 68, 200 - 229, 2004.

Abstract: We describe the general solution (α,β), where α=(α(s,x))s∈ ℂ and β=(β(s,x))s∈ ℂ are families of formal power series in C[[x]], of the two so-called cocycle equations

α(s+t,x)= α(s,x)α(t,π(s,x)),        s,t∈ ℂ (Co1)
β(s+t,x)= β(s,x)α(t,π(s,x)) +β(t,π(s,x)),         s,t∈ ℂ (Co2)
together with the boundary condition
α(0,x)=1,         β(0,x)=0, (B1)
where π=(π(s,x))s∈ ℂ is an iteration group in C[[x]]. Our method is based on the knowledge of the regular solutions of (Co1) and (Co2) and on a well-known and often used theorem concerning the algebraic relations between exponential functions and additive functions.
harald.fripertinger "at" uni-graz.at, October 12, 2017

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