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Rational dynamical zeta functions for birational transformations

Author

Listed:
  • Abarenkova, N.
  • Anglès d'Auriac, J.-Ch.
  • Boukraa, S.
  • Hassani, S.
  • Maillard, J.-M.

Abstract

We propose a conjecture for the exact expression of the unweighted dynamical zeta function for a family of birational transformations of two variables, depending on two parameters. This conjectured function is a simple rational expression with integer coefficients. This yields an algebraic value for the topological entropy. Furthermore, the generating function for the Arnold complexity is also conjectured to be a rational expression with integer coefficients with the same singularities as for the dynamical zeta function. This leads, at least in this example, to an equality between the Arnold complexity and the exponential of the topological entropy. We also give a semi-numerical method to effectively compute the Arnold complexity.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:phsmap:v:264:y:1999:i:1:p:264-293
    DOI: 10.1016/S0378-4371(98)00452-X
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
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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.

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