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New integrable cases of a Cremona transformation: a finite-order orbits analysis

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  • Boukraa, S.
  • Hassani, S.
  • Maillard, J.-M.

Abstract

We analyse the properties of a particular birational mapping of two variables (Cremona transformation) depending on two free parameters (ε and α), associated with the action of a discrete group of non-linear (birational) transformations on the entries of a q × q matrix. This mapping originates from the analysis of birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. It has been seen to yield weak chaos and integrability. We have found new integrable cases of this Cremona transformation, corresponding to the values of α = 0 when ε = 12, 13, + 1, besides the already known values ε = 0 and ε = −1, and also arbitrary α when ε = 0. For these cases, one has a foliation of the parameter space in elliptic curves. We give the equations of these elliptic curves. Based on this very example we show how one can find these integrability cases of the Cremona transformation and actually integrate it using a method based on the systematic study of the finite-order conditions of the Cremona transformation. The method is shown to be efficient and straightforward. The various integrability cases are revisited using many different representations of this very mapping (birational transformations, recursion in one variable, …).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:240:y:1997:i:3:p:586-621
    DOI: 10.1016/S0378-4371(97)00165-9
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    References listed on IDEAS

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    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. 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.
    3. Meyer, H. & Anglès d'Auriac, J.C. & Maillard, J-M. & Rollet, G., 1994. "Phase diagram of a six-state chiral Potts model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 209(1), pages 223-236.
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    Cited by:

    1. 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.
    2. 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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    1. 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.
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    4. 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.

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