Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON THE DYNAMICS OF BIRATIONAL MAPPINGS OF THE PLANE

  • Bedford, Eric (Department of Mathematics Indiana University Bloomington)
  • Published : 2003.05.06
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Abstract

In this paper we discuss how the dynamics of certain birational maps of the real plane may be studied using complex methods.

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References

  1. N. Abarenkova, J.-Ch. Angles d'Auriac, S. Boukraa, S. Hassani and J.-M. Mail-lard, Real Arnold complexity versus real topological entropy for birational trans-formations, J. Phys. A: Math. Gen. 33 (2000), 1465-1501. https://doi.org/10.1088/0305-4470/33/8/301
  2. N. Abarenkova, J.-Ch. Angles d'Auriac, S. Boukraa and J.-M. Maillard, Growth complexity spectrum of some discrete dynamical systems, Physica D 130 (1999), 27-42. https://doi.org/10.1016/S0167-2789(99)00014-7
  3. N. Abarenkova, J.-Ch. Angles d'Auriac, S. Boukraa and J.-M. Maillard, Real topological entropy versus metric entropy for birational measure-preserving transformations, Physica D 144 (2000), 387-433. https://doi.org/10.1016/S0167-2789(00)00079-8
  4. E. Bedford and J. Diller, Energy and invariant measure for birational surface maps, preprint.
  5. E. Bedford and J. Diller, Real and complex dynamics of a family of birational maps of the plane: The golden mean subshift, preprint.
  6. E. Bedford and J. Diller, in preparation
  7. E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$. VIII: Quasi-expansion, Amer. J. Math. 124 (2002), 221–271. https://doi.org/10.1353/ajm.2002.0008
  8. E. Bedford and J. Smillie, Real polynomial diffeomorphisms with maximal entropy: Tangencies, Math. DS/0103038.
  9. R. Devaney and Z. Nitecki, Shift automorphisms in the H´enon family, Comm. Math. Phys. 67 (1979), 48–137. https://doi.org/10.1007/BF01221362
  10. J. Diller, Dynamics of birational maps of $P^2$, Indiana U. Math. J. 45 (1996), 721–772.
  11. J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. of Math. 123 (2001), 1135–1169. https://doi.org/10.1353/ajm.2001.0038
  12. C. Favre, Points périodiques d'applications birationnelles de $P^2$, Ann. Inst. Fourier (Grenoble) 48 (1998), 999–1023.
  13. S. Friedland, Entropy of rational self-maps of projective varieties, in Dynamical Systems and Related Topics (Nagoya, 1990), pages 128–140. World Sci. Publishing, River Edge, NJ, 1991.
  14. J. Hietarinta and C. M. Viallet, Singularity confinement and chaos in discrete systems, Phys. Rev. Lett. 81 (1997), 325–328. https://doi.org/10.1103/PhysRevLett.81.325
  15. S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Univ. Press, 2000
  16. R. Oberste-Vorth, Complex horseshoes and the dynamics of mappings of two complex variables, Ph.D dissertation, Cornell Univ., Ithaca, NY, 1987
  17. T. Takenawa, A geometric approach to singularity confinement and algebraic entropy, J. Phys. A: Math. Gen. 34 (2001), L95–L102. https://doi.org/10.1088/0305-4470/34/10/103
  18. T. Takenawa, Algebraic entropy and the space of initial values for discrete dynamical systems, J. Phys. A: Math. Gen. 34 (2001), 10533–10545. https://doi.org/10.1088/0305-4470/34/48/317
  19. T. Takenawa, Discrete dynamical systems associated with root systems of indefinite type, Comm. Math. Phys. 224 (2001), 657–681. https://doi.org/10.1007/s002200100568

Cited by

  1. On the degree growth of birational mappings in higher dimension vol.14, pp.4, 2004, https://doi.org/10.1007/BF02922170
  2. On the complex dynamics of birational surface maps defined over number fields vol.0, pp.0, 2016, https://doi.org/10.1515/crelle-2015-0113