Toral automorphism ($T_{A}x=Ax(\mathrm{mod}{\mathbb{Z}}^n)$)

An integer matrix $A$ with determinant $\pm 1$ defines a toral automorphismon the real torus ${\mathbb{R}}^d/{\mathbb{Z}}^d$. Integer entries make the map independent of the chosen representative. The inverse matrix also has integer entries, so the transformation is invertible. It preserves normalized Haar measure.

The map is a mixing measure-preserving transformation exactly when no eigenvalue of $A$ is a root of unity. This can be proved with characters of a real torus $e_k(x)=e^{2\pi ik\cdot x}$, since $e_k\circ T_A=e_{A^{T}k}$. Orthogonality givesIf no eigenvalue is a root of unity, a nonzero integer frequency cannot repeat: a repetition would give $((A^T{)}^m-I)k=0$ for some $m>0$. It therefore eventually leaves every finite set of frequencies. The displayed correlations prove mixing first for trigonometric polynomials, then for square-integrable functions by approximation.

Conversely, a root-of-unity eigenvalue makes $(A^T{)}^m-I$ singular for some $m$. As an integer matrix it has a nonzero rational kernel vector, which can be scaled to a nonzero integer vector $k$. The nonconstant character $e_k$ is then invariant under $T_A^m$, obstructing mixing. The cat map is a mixing example, while the identity matrix gives an invertible measure-preserving map that is not mixing.