= Toral automorphism
{title2=$T_Ax=Ax\pmod{\mathbb Z^n}$}
An integer <matrix> with determinant $\pm1$ 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.
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