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Two-dimensional continued fractions. The simplest examples. (English. Russian original) Zbl 0883.11034
Proc. Steklov Inst. Math. 209, 124-144 (1995); translation from Tr. Mat. Inst. Steklova 209, 143-166 (1995).
Klein’s geometrical interpretation of ordinary continued fractions by the boundaries of convex hulls of points of the integer lattice in two-dimensional cones initiated the development of various concepts of $$(n+1)$$-dimensional continued fraction expansions, e.g. those by Voronoi and Minkowski. A central question is how to reconstruct the $$n$$-dimensional boundaries of such convex hulls from appropriate local characteristics and to judge, in particular, their periodicity. The author specifies a set of local $$\text{GL}_3(\mathbb{Z})$$ invariants that are sufficient for the unique reconstruction of the convex hull and the cones generated by the natural analog of the (one-dimensional) Fibonacci operator and some further operators which give rise to comparatively simple combinatorial structures.
For the entire collection see [Zbl 0863.00018].
Reviewer: G.Ramharter (Wien)

##### MSC:
 11J70 Continued fractions and generalizations