One generalization is to the theory of sails. If $A$ is an $ntimes n$ integer matrix whose eigenvalues are all real, positive, irrational and distinct, a collection of $n$ suitable eigenvectors spans a polyhedral cone which is invariant under $A$. The convex hull of the set of integer lattice points in this cone is a polyhedron, and the vertices of this polyhedron are the ``best'' integral approximations to the eigenvectors. Also see Arnold, e.g.
MR1704965 (2000h:11012)
Arnold, V. I.(RS-AOS)
Higher-dimensional continued fractions. (English, Russian summary)
J. Moser at 70 (Russian).
Regul. Chaotic Dyn. 3 (1998), no. 3, 10--17.
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