A *toral automorphism* is a map where . is *hyperbolic* if has no eigenvalue on the unit circle. has *unstable index 1* if has a single eigenvalue with modulus greater than one and if this eigenvalue has no algebraic multiplicity.

Observe that must be a *Pisot number*: it is an algebraic integer whose algebraic conjugates (ie, the other roots of , the smallest degree non-zero integer, monic polynomial such that ) must lie inside the open unit disk.

Note that the characteristic polynomial of must satisfy: with another integer monic polynomial with constant coefficient 1. Hence if is not a constant, then it would have a root of modulus at least 1, a contradiction. Hence, .

It is sometimes useful to know that one can ensure that all the eigenvalues of 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 are real. The latter can be ensured by a theorem of Pisot (see Theorem 5.2 of this book).

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