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Almost integrable mappings. (English) Zbl 0804.58045
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.

37K35Lie-Bäcklund and other transformations
14E05Rational and birational maps
14J50Automorphisms of surfaces and higher-dimensional varieties
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82C20Dynamic lattice systems and systems on graphs
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