Constructing a Fractional Linear Map

Question

I am working on a practice prelim question:

"Construct a nonlinear fractional map $\phi(z) = \frac{az+b}{cz+d}$ with $c \ne 0$ such that $\phi(\phi(\phi(z))) = z$. I feel like I just need to take the composition and just work it out to solve for the constants $a,b,c$ and $d$. Is this the right idea? Or is there a short cut?

Answer 1

Your method will work just fine. One suggestion to ease the calculations is to use the fact that composition of $\phi$ is equivalent to multiplying $\begin{pmatrix} a & b\\ c & d\end{pmatrix}$ by itself. Hence you need to work out $\begin{pmatrix} a & b\\ c & d\end{pmatrix}^3=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}.$

Answer 2

Do you know the connection between fractional linear transformations and matrices? The equation is true if the matrix $M = \pmatrix{a & b\cr c & d\cr}$ satisfies $M^3 = I$.

Answer 3

Hint: write that as $\phi \circ \phi = \phi^{-1}$. Solve $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}^2 = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ instead of computing the third power of the matrix.

Answer 4

Why don’t you just take $z\mapsto\omega z$, where $\omega$ is a primitive cube root of unity, and conjugate your linear map by a fractional linear that’s so nonlinear that the result is nonlinear? You’re surely looking for a $120^\circ$-rotation about some point in the plane.

EDIT: That was too complicated. Just take a real fractional-linear that maps $0\mapsto1\mapsto\infty\mapsto0$.