Annals of Combinatorics 3 (1999) 131-158

Functional Relations in Lattice Statistical Mechanics,Enumerative Combinatorics, and Discrete Dynamical Systems

J.-Ch. Anglè s d'Auriac1, S. Boukraa2, and J.-M. Maillard3

1Centre de Recherches sur les Très Basses Températures, B.P. 166, F-38042 Grenoble, France

2Institut d' Aé ronautique,Université de Blida, BP 270, Blida, Algeria

3LPTHE, Tour 16, 1er ètage, 4 Place Jussieu,75252 Paris Cedex, France

Received November 11, 1998

AMS Subject Classification: 82A68, 82A69, 14J50, 16A46, 16A24

Abstract.We recall some non-trivial, non-linear 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 singled out 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.

Keywords: functional relations, inversion relations, Yang-Baxter equations, rational dynamical zeta functions, discrete dynamical systems, birational mappings, Cremona transformations, analyticity assumptions

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