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

An integer matrix with determinant $\pm 1$ induces an invertible transformation of the real torus, preserving normalized Haar measure. It is a mixing measure-preserving transformation exactly when none of its eigenvalues is a root of unity. Indeed Fourier frequencies evolve by $A^T$: a nonzero integer frequency cannot revisit a fixed frequency unless some power has eigenvalue one. Conversely a root of unity gives a nonzero integer vector fixed by a power of $A^T$, and its nonconstant character obstructs mixing.