Understanding Abel/Schroeder with matrix-expression

[bo198214]

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.