Regular iteration using matrix-Jordan-form

[Gottfried]

Maybe I read Gottfried's post too fast , but what is the difference/benefit between using Jordan form and the matrix-logarithm ?

Maybe its in the links , I did not read them yet.
(...)

Well, not in the "links" but in the "connections" ;-) . If one wants to describe the Carleman-approach for the iterated exponential (and other iterations) then it is of interest to see as many connections to other parts in mathematics as possible to widen the view on the problem (remember there is still a lot of things missing, let us only remember the uniqueness problem). It is like collecting related information for some encyclopedic resource like wikipedia or the tetration-wiki, helpful for any newly interested collaborators.
The new information by the Jordan-decomposition is, that we introduce - based on the same logic as in the diagonalization - the use of iteration-series: one thing I expect/I hope to lead (in some future) to a better evaluable series than the otorious "zero-convergence-radius power series" for the fractional iterates, and also have a well known formalism which also connects the widely known binomial composition via Newton-series with the diagonalization-approach.
I think it's just worth to have that connections explicite.

Gottfried