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Functional relations in lattice statistical mechanics, enumerative combinatorics and discrete dynamical systems. (English) Zbl 1055.82505
Summary: We recall some non-trivial, nonlinear functional relations appearing in various domains of mathematics and physics, such as lattice statistical mechanics, quantum mechanics, or enumerative combinatorics. We focus, more particularly, on the analyticity properties of the solutions of these functional relations. We then consider discrete dynamical systems corresponding to birational transformations. The rational expressions for dynamical zeta functions obtained for a particular two-dimensional birational mapping depending on two parameters, are recalled, as well as some non-trivial functional relations satisfied by these dynamical zeta functions. We finally give some functional equations corresponding to some special orbits of this two-dimensional birational mapping for particular values of the two parameters. This example shows that functional equations associated with curves, for real values of the variables, are actually compatible with a chaotic dynamical system.
MSC:
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
14E05Rational and birational maps
37C30Functional analytic techniques in dynamical systems
39B05General theory of functional equations
82B23Exactly solvable models; Bethe ansatz
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