 Cat map

The cat map is the area-preserving toral automorphism induced by the integer matrix $A=\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$. Its eigenvalues are $(3\pm \sqrt{5})/2$, giving one expanding and one contracting eigendirection. Cutting its image parallelogram by the integer grid and translating the pieces modulo one gives a concrete geometric construction on the unit square.