iteration of fractional linear functions

[Gottfried]

The interesting thing is that fractional linear functions, i.e. mappings of the form
\(
h_A(x)=\frac{ax+b}{cx+d}
\)
can be represented with help of the matrix
\( A=\begin{pmatrix}a &b\\c& d\end{pmatrix} \).
as follows.

The composition of these maps corresponds to the composition of their matrices!

\( h_{AB}=h_A \circ h_B \).
We know this phenomenon from the Carleman matrix! But \( A \) is *not* the Carleman matrix of \( h_A \).

The representation is only unique up to a fraction extension constant:
\( h_A = \operatorname{id}\Leftrightarrow A = \alpha I \) for some complex \( \alpha \).

Thatswhy we have here a natural way of fractionally iterating these fractional linear functions, i.e. via matrix powers.

Hi Henryk -

quite interesting; though it didn't come to my attention when I fiddled a bit with functions like this.

I vaguely remember that Gottfried posted somewhen a link to a thread on sci.math that also discusses the iteration of fractional linear functions (so if you still know, Gottfried, perhaps you can repost it here).

I'm sorry, I have a vague idea of such postings but no true memory. Perhaps if we look for posts of alain verghote, or we may email him personally. He seems to have experiences with/lists of several of such type of functions.
Alain Verghote alainverghote (att) gmail(.)com
I'm not much with math this days, so better someone else gets in contact with him. He's a friendly person who likes to exchange about that subject.

Gottfried