I have just read a paper about fractional linear functions written by Justin Lanier. I have tried to prove some observations he left in page 13. Here is the statement I want to prove:
Given function $f(x)=\frac{ax+b}{cx+d}$ ,then
$$f^{\circ n}(x)=x \iff \frac{a+d}{2\sqrt{ad-bc}}=\cos(\frac{k\pi}{n}), 1\leq k\leq n-1$$
The cycle period is $\gcd(k,n)$
$f^{\circ n}(x)$ denotes $n$-th iterate of function $f(x)$
I will sketch my proof here: $f^{\circ n}(x)=\frac{A_nx+B_n}{C_nx+D_n}$ and the following system is satisfied:
$$
\begin{cases}
R_n=\frac{A_n-D_n}{a-d}\\
A_{n+1}=aA_n+bcR_n\\
D_{n+1}=dD_n+bcR_n\\
B_n=bR_n\\
C_n=cR_n\\
A_0=B_0=1
\end{cases}
$$
thus $f^{\circ n}(x)=x\iff R_n=0$. Sequences $A_n,B_n$ satisfy a system of linear recurrence relations, so with the help of Linear algebra, the closed form of $R_n$ is:
$$R_n=\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2} $$ $\lambda_1,\lambda_2$ are 2 roots of the equation $\lambda^2-(a+d)\lambda+(ad-bc)=0$. It is certain that equation $R_n=0$ yields the result.
What I want to know:
- Is my idea correct? Honestly, I'm worried that the statement is not complete. I checked a lot of cases and the computation seems to agree.
- If you know any other papers writing about this problem, please let me know. I couldn't find them.