The real hyperbolic fixed points of $\mathrm{SL}\_2(\mathbb{Z})$ are the points $x\in\mathbb R\smallsetminus\mathbb{Q}$ with
$$
\frac{ax+b}{cx+d}=x
$$
for some $\left(\begin{array}{2} a&b\\ c&d\end{array}\right)\in\mathrm{SL}\_2(\mathbb{Z})$. These are special quadratic numbers and I wonder, what their properties are.
Do they have a name? Arithmetic meaning?



Hyperbolic fixed points of SL(2,Z)
Am I missing anything?


Any fixed point $x$ is a root of quadratic equation $px^2+qx+r=0$ with $p,q,r\in\mathbb Z$, where $D=q^2-4pr$ is not a square; we assume that $(p,q,r)=1$. This $x$ must also satisfy $cx^2+(d-a)x-b=0$, which yields $c=\nu p$, $b=-\nu r$, $d=\nu q+a$ for an integer $\nu$. So we need
$$
 1=\left|\matrix{a& -\nu r\\\nu p& \nu q+a}\right|=a^2+\nu q a+\nu^2pr
$$
to be solvable in integer $a$. This is so iff $(\nu q)^2-4(\nu^2pr-1)=\nu^2D+4$ is a square.


But, since $D$ is not a square, a Pell's equation $n^2-D\nu^2=4$ has a non-trivial solution $(n,\nu)$. Starting at that, we get
$$
 a=\frac{n-\nu q}2, \quad b=-\nu r, \quad c=\nu p, \quad d=\nu q+a
$$
which form a desired element in $\mathrm{SL}\_2(\mathbb Z)$.


So *every* quadratic irrational (in $\mathbb R$ or in $\mathbb C$, depending of what you are transforming) is a fixed point, and this is a complete description.


