Since the eigenvectors have rationally independent components both the eigenspaces densely cover the torus. Arnold's cat map is a particularly well-known example of a hyperbolic toral automorphism, which is an automorphism of a torus given by a square unimodular matrix having no eigenvalues of absolute value 1.
Since eigenvectors have rationally independent components, the eigenspaces completely cover the torus. Arnold's cat map is a well-known example of a hyperbolic toral automorphism, which is an automorphism of a torus given by a square unimodular matrix with no eigenvalues equal to 1.