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Zbl 0804.58045
Boukraa, S.; Maillard, J.-M.; Rollet, G.
Almost integrable mappings. (English)
[J] Int. J. Mod. Phys. B 8, No.1-2, 137-174 (1994). ISSN 0217-9792

Summary: We analyze birational transformations obtained from very simple algebraic calculations, namely taking the inverse of $q \times q$ matrices and permuting some of the entries of these matrices. We concentrate on $4 \times 4$ matrices and elementary transpositions of two entries. This analysis brings out six classes of birational transformations. Three classes correspond to integrable mappings, their iteration yielding elliptic curves. Generically, the iterations corresponding to the three other classes are included in higher dimensional non-trivial algebraic varieties. Nevertheless some orbits of the parameter space lie on (transcendental) curves. These transformations act on fifteen (or $q\sp 2 - 1$) variables, however one can associate to them remarkably simple non- linear recurrences bearing on a single variable. The study of these last recurrences gives a complementary understanding of these amazingly regular non-integrable mappings, which could provide interesting tools to analyze weak chaos.

MSC 2000:: *37K35 Lie-Bäcklund and other transformations
14E05 Birational correspondences
14J50 Automorphisms of surfaces and higher-dimensional varieties
82B20 Lattice systems
82C20 Dynamic lattice systems

Keywords: lattice models; statistical mechanics; birational transformations; integrable mappings; iterations; algebraic varieties

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