Let be the affine reflection described as an operation with complex numbers : $$s_\beta,_\nu : z \mapsto \nu + \overline{\beta z},$$ where $z, \nu \in \Bbb C$ and $\beta \in \Bbb C^1 = \{x+iy \ \mathrm{such \ that} \ x^2 + y^2 = 1; x,y \in \Bbb R\}$
I want to show that if $\nu \neq 0 \ \mathrm{and} \ \beta \neq -\frac{\overline{\nu^2}}{\lvert \nu\rvert^2} $, then $s_\beta,_\nu$ is of infinite order. I've already proved (after using an induction reasoning on the form of the k-th composition) that for an even number of compositions of $s_\beta,_\nu$, you can't get the identity operation. But I'm struggling to prove it for an odd number of compositions.
I have that :
$s_\beta,_\nu^k(z) = \frac{k}{2}\nu + \frac{k}{2}\overline{\beta\nu} + z$, if $k$ is even; and $s_\beta,_\nu^k(z) = \frac{k+1}{2}\nu + \frac{k-1}{2}\overline{\beta\nu} + \overline{\beta z}$, if $k$ is odd; $k \geq 2, k \in \Bbb N$.
Thanks.
If $k$ is odd, $s_\beta,_\nu^k(z) = \frac{k+1}{2}\nu + \frac{k-1}{2}\overline{\beta\nu} + \overline{\beta z}$ reverses orientation of the plane.