How do we set about drawing a toral automorphism as in figure 5.1 in the picture above.
How do we know where the points highlighted in yellow are?
What happens if the eigenvalue (Im guessing some of these lines are eigenlines) has multiplicity 2? Do we just use the generalised eigenvector as the second eigenvector?
To add a few details: to get an automorphism you'd better be able to undo the map $T$, and to be able to map the image of the square back to the torus $T$ had better map integer lattice points to integer lattice points. If $T$ is also linear, as we have here, then these conditions are equivalent to requiring $T\in GL_2(\mathbb{Z})$, that is, $T$ is represented by an invertible integer matrix. Then to draw a picture like this you simply apply the matrix to the square; but since the sides of the square are the standard basis vectors, you know the legs of the resulting parallelogram that begin at the origin will just be the columns of the matrix of $T$.