= Toral automorphism
{title2=$T_Ax=Ax\pmod{\mathbb Z^n}$}
An integer <matrix> $A$ with <determinant> $\pm1$ defines a <toral automorphism>
$$
T_Ax=Ax\pmod{\mathbb Z^d}
$$
on 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> $e_k(x)=e^{2\pi i k\cdot x}$, since $e_k\circ T_A=e_{A^Tk}$. Orthogonality gives
$$
\int e_k(T_A^nx)\overline{e_\ell(x)}\,dx=
\begin{cases}1,&(A^T)^nk=\ell,\\0,&\text{otherwise}.\end{cases}
$$
If 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.
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