Dynamical Classification of a Family of Birational Maps of C 2 via Algebraic Entropy

• Zafar, Sundus ^[1] ; Cima, Anna ^[1]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 18, Nº 2, 2019, págs. 631-652
• Idioma: inglés
• DOI: 10.1007/s12346-018-0304-1
• Enlaces
  • Texto completo
• Resumen

  • This work dynamically classifies a 9-parametric family of birational maps f:C2→C2. From the sequence of the degrees dn of the iterates of f,  we find the dynamical degree δ(f) of f. We identify when dn grows periodically, linearly, quadratically or exponentially. The considered family includes the birational maps studied by Bedford and Kim (Mich Math J 54:647–670, 2006) as one of its subfamilies.

• Referencias bibliográficas
  • 1. Angles, J.C., Maillard, J.M., Viallet, C.: On the complexity of some birational transformations. J. Phys. A Math. Gen. 39, 36413654 (2006)
  • 2. Barth, W., Hulek, K., Peters, C., Ven, A.: Compact Complex Surfaces, vol. 4. Springer, Berlin (2004)
  • 3. Bedford, E., Kim, K.: On the degree growth of birational mappings in higher dimension. J. Geom. Anal. 14, 567–596 (2004)
  • 4. Bedford, E., Kim, K.: Periodicities in linear fractional recurrences: degree growth of birational surface maps. Mich. Math. J. 54, 647–670...
  • 5. Bedford, E., Kim, K.: The dynamical degrees of a mapping. In: Proceedings of the Workshop Future Directions in Difference Equations, Colecc....
  • 6. Bellon, M.P., Viallet, C.M.: Algebraic entropy. Comm. Math. Phys. 204(2), 425–437 (1999)
  • 7. Blanc, J., Deserti, J.: Degree Growth of Birational Maps of the Plane arXiv:1109.6810 [math.AG] (2012)
  • 8. Blanc, J., Cantat, S.: Dynamical Degree of Birational Transformations of Projective Surfaces arXiv:1307.0361 [math.AG] (2013)
  • 9. Cima, A., Zafar, S.: Zero Entropy for Some Birational Maps of C2, arXiv:1704.07108
  • 10. Cima, A., Zafar, S.: Invariant Fibrations for Some Birational Maps of C2, arXiv:1702.00959
  • 11. Cima, A., Zafar, S.: Integrability and algebraic entropy of k-periodic non-autonomous Lyness recurrences. J. Math. Anal. Appl. 413, 20–34...
  • 12. Cima, A., Gasull, A., Mañosa, V.: Dynamics of some rational discrete dynamical systems via invariants. J. Bifurc. Chaos 16(3), 631–645...
  • 13. Cima, A., Gasull, A., Mañosa, V.: On 2- and 3-periodic Lyness difference equations. J. Differ. Equ. Appl. 18(5), 849–864 (2012)
  • 14. Cima, A., Gasull, A., Mañosa, V.: Non-autonomous 2-periodic Gumovski–Mira difference equations. Int. J. Bifurc. Chaos Appl. Sci. Eng....
  • 15. Devault, R., Schultz, S.W.: On the dynamics of xn+1 = β xn+γ xn−1 B xn+D xn−2 . Commun. Appl. Nonlinear Anal. 12, 3540...
  • 16. Diller, J.: Dynamics of birational maps of PC2. Indiana Univ. Math. J. 45(3), 721–772 (1996)
  • 17. Diller, J., Favre, C.: Dynamics of bimeromorphic maps of surfaces. Am. J. Math. 123, 1135–1169 (2001)
  • 18. Fornaes, J.-E., Sibony, N.: Complex dynamics in higherdimension. In: II, Modern Methods in Complex Analysis (Princeton, NJ, 1992), 135182,...
  • 19. Ladas, G.: On the rational recursive sequence xn+1 = a+β xn+xn−1 A+B xn+C xn−1 . J. Differ. Equ. Appl. 1, 317321 (1995)
  • 20. Ladas, G., Kulenovic, M., Prokup, N.: On the recursive sequence xn+1 = a xn+b xn−1 A+xn . J. Differ. Equ. Appl. 6, 563–576...
  • 21. Ladas, G., Gibbons, C.H., Kulenovic, M.R.S.: On the rational recursive sequence xn+1 = α+β xn+γ xn−1 A+B xn . In:...
  • 22. Ladas, G., Kulenovic, M.R.S., Martins, L.F., Rodrigues, I.W.: On the dynamics of xn+1 = a+b xn A+B xn+C xn−1 . Facts...
  • 23. Pettigrew, J., Roberts, J.A.G.: Characterizing singular curves in parametrized families of biquadratics. J. Phys. A 41((11)), 115203 (2008)....
  • 24. Roberts, J.A.G.: Order and symmetry in birational difference equations and their signatures over finite phase spaces. In: Proceedings...
  • 25. Yomdin, Y.: Volume growth and entropy. Isr. J. Math. 57, 285–300 (1987)
  • 26. Zafar, S.: Dynamical Classification of Some Birational Maps of C2. Ph.D dissertation, Universitat Autónoma de Barcelona, Spain (2014)
  • 27. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive sequence xn+1 = a xn+b xn−k cxn−d xn−k . Commun. Appl. Nonlinear...