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Integrable mappings and polynomial growth

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  • Boukraa, S.
  • Maillard, J-M.
  • Rollet, G.

Abstract

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on q × q matrices: the inversion of the q × q matrix and an (involutive) permutation of the entries of the matrix. We concentrate on the case where these permutations are elementary transpositions of two entries. In this case the birational transformations fall into six different classes. For each class we analyze the factorization properties of the iteration of these transformations. These factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties. Some mappings yield a polynomial growth of the complexity of the iterations. For three classes the successive iterates, for q = 4, actually lie on elliptic curves. This analysis also provides examples of integrable mappings in arbitrary dimension, even infinite. Moreover, for two classes, the homogeneous polynomials are shown to satisfy non trivial non-linear recurrences. The relations between factorizations of the iterations, the existence of recurrences on one or several variables, as well as the integrability of the mappings are analyzed.

Suggested Citation

  • Boukraa, S. & Maillard, J-M. & Rollet, G., 1994. "Integrable mappings and polynomial growth," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 209(1), pages 162-222.
    • Handle: RePEc:eee:phsmap:v:209:y:1994:i:1:p:162-222
    DOI: 10.1016/0378-4371(94)90055-8
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    Cited by:

    1. Boukraa, S & Maillard, J-M, 1995. "Factorization properties of birational mappings," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 220(3), pages 403-470.
    2. Abarenkova, N. & Anglès d'Auriac, J.-C. & Maillard, J.-M., 1997. "More integrable birational mappings," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 237(1), pages 123-134.
    3. Abarenkova, N. & Anglès d'Auriac, J.-Ch. & Boukraa, S. & Hassani, S. & Maillard, J.-M., 2000. "Real Arnold complexity versus real topological entropy for a one-parameter-dependent two-dimensional birational transformation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 281(1), pages 151-172.
    4. Boukraa, S. & Hassani, S. & Maillard, J.-M., 1997. "New integrable cases of a Cremona transformation: a finite-order orbits analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 240(3), pages 586-621.
    5. Abarenkova, N. & Anglès d'Auriac, J.-Ch. & Boukraa, S. & Hassani, S. & Maillard, J.-M., 1999. "Rational dynamical zeta functions for birational transformations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 264(1), pages 264-293.

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