Understanding Abel/Schroeder with matrix-expression

[Gottfried]

Anyway, what is still surprising, is, that the \( \sigma \)-function seems to have an exponential characteristic. I'd expected something near a logarithmic characteristic instead.

Hm, the logarithm of a Schroeder function is an Abel function:
\( \sigma(f(x))=u\sigma(x) \)
\( (\log_u\circ \sigma)(f(x))=1+(\log_u\circ\sigma)(x) \)

So the logarithm should look like an slog.

Yes; may be I didn't see the equality since I had always a different powerseries due to a different fixpoint-shift.

In the light of my last msg it seems to me, that possibly the application of the Schroeder-function has a benefit over the Abel-function because we do not explicitely deal with logarithms; this is of significance for all infinite heights and complex heights (log(0),log(-x)), which occurs if x is "out-of-bounds" for b in the range e^(-e) < b < e^(1/e) and x>t .
I didn't work with the slog up to now, so I can't say anything about this.
[update] Concerning the expectation of a "logarithmic characteristic": I meant this, because x is an iterated exponential which is converted into a product. This conversion to a lower operation is usually connected with some logarithm-function. So I thought, the Schröder-function would be roughly similar to a logarithmic function... [/update]