$SL(2,\mathbb{R})$ in fusion reactors.

The Poincaré map (yellow arrow) is defined by following a magnetic field line (purple) around the torus until it comes back to the same initial cross-section. Larger image. Images courtesy of Christopher Smiet.

The three stable configurations for a $2\times 2$ matrix $M$ of determinant 1 depend on its trace $\mbox{Tr}(M)$. Larger image. From left to right: $M$ is alternating hyperbolic if $\mbox{Tr}(M)<-2$; $M$ is elliptic if $-2<\mbox{Tr}(M)<2$; $M$ is hyperbolic if $2 <\mbox{Tr}(M)$. Here the two ends of a colored arrow correspond to a point ${\bf x}$ and its image $M{\bf x}$.