06/14/2009, 05:16 PM
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.
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).
Further investigation shows that indeed the eigenvalues of the matrix are non-real if and only if the function has non-real fixed points. As we know the fractional iteration via matrix powers is given by linear combinations of powers of the eigenvalues of the matrix.
So it seems that the iteration via matrix powers is linked to the iteration at the two fixed points (which is also real if the fixed point is real).
So now my question to the audience of this forum. We have only one matrix power iteration but we have the regular iteration iteration at the two fixed points of \( h_A \). So the question is how do they relate. I make a poll and ask you about your opinions (I was just too lazy yet to verify it theoretically which should not be too difficult, but at least that way we can check how good our intuition is.)
\(
h_A(x)=\frac{ax+b}{cx+d}
\)
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.
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).
Further investigation shows that indeed the eigenvalues of the matrix are non-real if and only if the function has non-real fixed points. As we know the fractional iteration via matrix powers is given by linear combinations of powers of the eigenvalues of the matrix.
So it seems that the iteration via matrix powers is linked to the iteration at the two fixed points (which is also real if the fixed point is real).
So now my question to the audience of this forum. We have only one matrix power iteration but we have the regular iteration iteration at the two fixed points of \( h_A \). So the question is how do they relate. I make a poll and ask you about your opinions (I was just too lazy yet to verify it theoretically which should not be too difficult, but at least that way we can check how good our intuition is.)