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toral automorphism $f:\mathbb T^d\to\mathbb T^d$ is a map $x\mapsto Ax\mod\mathbb Z^d$ where $A\in GL(d,\mathbb Z)$. $f$ is hyperbolic if $A$ has no eigenvalue on the unit circle. $f$ has unstable index 1 if $A$ has a single eigenvalue $\lambda$ with modulus greater than one and if this eigenvalue has no algebraic multiplicity.
Observe that $\lambda$ must be a Pisot number: it is an algebraic integer whose algebraic conjugates (ie, the other roots of $P_\lambda$, the smallest degree non-zero integer, monic polynomial such that  $P_\lambda(\lambda)=0$) must lie inside the open unit disk.
Note that the characteristic polynomial of $A$ must satisfy: $\chi_A=P_\lambda\cdot Q$ with $Q$ another integer monic polynomial with constant coefficient 1. Hence if $Q$ is not a constant, then it would have a root of modulus at least 1, a contradiction. Hence, $\chi_A=P_\lambda$.
It is sometimes useful to know that one can ensure that all the eigenvalues of $A$ are simple and have pairwise distinct moduli. By an observation of C. Smyth (Advanced Problem 5931, Amer. Math. Monthly 82 (1975), p.86) this occurs whenever all the roots of $P_\lambda$ are real. The latter can be ensured by a theorem of Pisot (see Theorem 5.2 of this book).