Maximal families of commuting automorphisms and the differential equation of Aczél-JabotinskyKnown results and open problemsPrelinear equationsExplicit solutions of the translation equation in formal power series rings

Explicit solutions of the translation equation in formal power series rings

Let F be an order preserving automorphism of the ring of formal power series C[[x1,...,xn]] (or of locally analytic diffeomorphisms of C) such that F | C= id C. (Let the group of these automorphisms be called G.) Does there exist a one parametric family (Ft)tÎC, FtÎG, which fulfills
Ft o Fs=Ft+s,    t,sÎC  (T)
and
F1=F ?  (E)
If so, then (Ft)tÎC is called iteration group (or embedding) of F. Depending on the relations between the coefficient functions of Ft(x)=F(t,x)=A(t)+P(t,x) and the group parameter t we call them analytic or continuous iterations or iterations without any regularity conditions. The iteration problem has its roots in analytic mechanics ([6][43]). The functional equation (T) is the well known translation equation and (E) is the embedding condition.

The iteration problem ((T),(E)) under various regularity conditions was solved by L. Reich and J. Schwaiger by using semicanonical forms with respect to conjugation of FÎG. In other words this approach was based on solutions of the generalized Schröder equation

T o N=F o T  (gS)
for N,TÎG, under certain assumptions on the structure of the (simplified) semicanonical form N. (Cf. [28][30][32][29].)

For example we want to present a criterion for the existence of an analytic iteration of F(x)=Ax+P(x) for given logarithms L=(l1,...,ln) of the eigenvalues r1,...,rn of A: F has an analytic iteration with respect to L (i.e. the eigenvalues of Ft are of the form el1t,...,elnt, for tÎC) if and only if F has a smooth normal form with respect to L. (Cf. [28].)

Proof techniques and constructions of iteration groups are based on methods of formal differential systems, difference equations, inhomogeneous Cauchy-equations and "complete linearization", which dates back to Erdös and Jabotinsky.

These criteria proved to be the proper base for further research in the topics of the iteration problem.

The translation equation (T) plays an important role in several branches of mathematics. Many publications describe general and special solutions of it. (Cf. [2][25].) Several facts are known about the solutions of

F(t,F(s,x))=F(t+s,x)    t,sÎC  (T)
if Ft=F(t,.)ÎG. For n=1 L. Reich described in [31] the structure of the coefficient functions. And he showed that they can be derived from the structure of the "analytic" solutions of (T) (i.e. solutions with analytic coefficient functions). In the general case it is known that the coefficients of F(t,x) are polynomials in additive and generalized exponential functions. The following problems are still open:
  1. Which of these additive functions may be chosen linearly independent and which of these exponential functions may be chosen multiplicativly independent? And where (at which indices) do they occur in the coefficient functions of Ft?
  2. Is it possible to derive the structure of the coefficients of F(t,x) from the analytic solutions of (T)? How to describe the analytic solutions?
harald.fripertinger@kfunigraz.ac.at,
last changed: February 9, 2001

Maximal families of commuting automorphisms and the differential equation of Aczél-JabotinskyKnown results and open problemsPrelinear equationsExplicit solutions of the translation equation in formal power series rings