Visualizing toral automorphisms (Q1209808)

The group of continuous automorphisms on the torus is isomorphic to \(\text{GL}(2,\mathbb{Z})\). Picture a torus constructed by joining two tubes of different colors. Figures show the mixing of the colors as the automorphism \({{2\;1}\choose {1\;1}}\) is iterated. A variety of properties of toral automorphisms are described and how to visualize them geometrically. The interested reader is encouraged to look at the pictures, they are worth more than a thousand words.