Linear functional equations and iteration theoryKnown results and open problemsExplicit solutions of the translation equation in formal power series ringsMaximal families of commuting automorphisms and the differential equation of Aczél-Jabotinsky

Maximal families of commuting automorphisms and the differential equation of Aczél-Jabotinsky

For n=1 (cf. [33][34][35][36][37]) there is a close connection between maximal families of commuting automorphisms of C[[x1,...,xn]] and iteration theory: Each maximal family F=(Fi)iÎI of FiÎG (i.e. Fi o Fj=Fj o Fi, i,jÎI) is the set of solutions of exactly one Aczél-Jabotinsky differential equation
H(x)(df)/(dx)=(H o f)(x),   (AJ)
for a given H(x)ÎC[[x]], ordH³1, where the solution f has to fulfill ordf=1. Since there is a singularity at x=0 the usual existence and uniqueness theorems cannot be applied.

Conversely the set of solutions of any equation (AJ) is a maximal family of commuting series. Moreover each such family is an iteration group (Ft)tÎC or it contains an iteration group (as a subgroup of finite index).

For n>1 no similar results are known about the automorphism of the rings C[[x1,...,xn]], whereas results exist in other situations. (Cf. [1][12][14][13].) As a matter of fact for n>1 each analytic iteration group (Ft)tÎC consists of solutions of a system of partial differential equations

(f)/(x)H(x)=(H o f)(x),   (AJn)
where
H(x)=(F(t,x))/(t) | t=0
and (f)/(x) is the Jacobi matrix of the automorphism fÎG. Solving the following problems for n>1 is an interesting and difficult problem:
  1. Description of the solvability and the solutions of (AJn).
  2. Describing the connection between maximal families of commuting series and the Aczél-Jabotinsky equations. Finding further differential equations or functional equations which together with (AJ) characterize the maximal families of commuting automorphisms.
  3. Iteration groups as subgroups of maximal families.
Even results just for the case n=2 would represent important progress.
harald.fripertinger@kfunigraz.ac.at,
last changed: February 9, 2001

Linear functional equations and iteration theoryKnown results and open problemsExplicit solutions of the translation equation in formal power series ringsMaximal families of commuting automorphisms and the differential equation of Aczél-Jabotinsky