 Hyperbolic toral automorphism

A hyperbolic toral automorphism is a toral automorphism $T_{A}x=Ax(\mathrm{mod}{\mathbb{Z}}^d)$ whose matrix has no eigenvalue on the unit circle. Here $A\in \mathrm{GL}(d,\mathbb{Z})$, so its determinant is $\pm 1$. Invertibility merely over the real numbers would not suffice: multiplication by two on a circle is a many-to-one endomorphism, not an automorphism.

The real stable and unstable subspaces come from the generalized eigenspaces with eigenvalue modulus less than and greater than one. They give the same splitting at every point of the real torus. A Jordan block can add a polynomial factor to the powers of its eigenvalue; choosing a slightly weaker exponential rate absorbs that factor into a constant. Thus contraction forward on the stable subspace and backward on the unstable subspace is uniform over the whole torus, proving that the map is an Anosov diffeomorphism.

For the cat map, $A=\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$ has eigenvalues $(3\pm \sqrt{5})/2$. One is inside and the other outside the unit circle, and their product is one. The toral automorphism mixing criterion applies as well, since a root of unity would lie on that circle.