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Publications mathématiques

, Volume 105, Issue 1, pp 49–89 | Cite as

Dynamics on blowups of the projective plane

  • Curtis T. McMullen
Article

Keywords

Projective Plane Weyl Group Coxeter Group Rational Surface Positive Entropy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes II, Ann. Math., 88 (1968), 451–491.CrossRefMathSciNetGoogle Scholar
  2. 2.
    G. Bastien and M. Rogalski, Global behavior of the solutions of Lyness’ difference equation u n+2 u n=u n+1+a, J. Difference Equ. Appl., 10 (2004), 977–1003.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Linear fractional recurrences, Preprint, 2006.Google Scholar
  4. 4.
    E. Bedford and K. Kim, Periodicities in linear fractional recurrences: Degree growth of birational surface maps, Preprint, 2005.Google Scholar
  5. 5.
    E. Bedford and J. Diller, Energy and invariant measures for birational surface maps, Duke Math. J., 128 (2005), 331–368.Google Scholar
  6. 6.
    Y. Bilu, Limit distribution of small points on algebraic tori, Duke Math. J., 89 (1997), 465–476.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R. E. Borcherds, The Leech Lattice and Other Lattice, Thesis, Trinity College, Cambridge, 1984.Google Scholar
  8. 8.
    N. Bourbaki, Groupes et algèbres de Lie, Chap. IV–VI, Hermann, 1968; Masson, 1981.Google Scholar
  9. 9.
    S. Cantat, Dynamique des automorphismes des surfaces projectives complexes, C. R. Acad. Sci., Paris, Sér. I, Math., 328 (1999), 901–906.zbMATHMathSciNetGoogle Scholar
  10. 10.
    S. Cantat, Dynamique des automorphismes des surfaces K3, Acta Math., 187 (2001), 1–57.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. B. Coble, Algebraic Geometry and Theta Functions, Colloquium Publications, vol. 10, Amer. Math. Soc., 1961.Google Scholar
  12. 12.
    J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups. Springer, 1999.Google Scholar
  13. 13.
    I. Dolgachev and D. Ortland, Point Sets in Projective Spaces and Theta Functions. Astérisque, vol. 165, 1988.Google Scholar
  14. 14.
    I. Dolgachev and D.-Q. Zhang, Coble rational surfaces, Amer. J. Math., 123 (2001), 79–114.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. Dujardin, Laminar currents and birational dynamics, Duke Math. J., 131 (2006), 219–247.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    V. Flammang, M. Grandcolas, and G. Rhin, Small Salem numbers, in Number Theory in Progress, vol. I, pp. 165–168, de Gruyter, 1999.Google Scholar
  17. 17.
    M. Gizatullin, Rational G-surfaces, Math. USSR Izv., 16 (1981), 103–134.zbMATHCrossRefGoogle Scholar
  18. 18.
    P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Interscience, 1978.Google Scholar
  19. 19.
    M. Gromov, On the entropy of holomorphic maps, Enseign. Math., 49 (2003), 217–235.zbMATHMathSciNetGoogle Scholar
  20. 20.
    B. Harbourne, Blowings-up of P2 and their blowings-down, Duke Math. J., 52 (1985), 129–148.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    B. Harbourne, Automorphisms of K3-like rational surfaces, in Algebraic Geometry, Bowdoin, 1985, Proc. Symp. Pure Math., vol. 48, pp. 17–28. Amer. Math. Soc., 1987.Google Scholar
  22. 22.
    B. Harbourne, Rational surfaces with infinite automorphism group and no antipluricanonical curve, Proc. Amer. Math. Soc., 99 (1987), 409–414.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    B. Harbourne, Iterated blow-ups and moduli for rational surfaces, in Algebraic Geometry (Sundance, UT, 1986), Lect. Notes Math., vol. 1311, pp. 101–117, Springer, 1988.Google Scholar
  24. 24.
    J. Harris and I. Morrison, Moduli of Curves, Springer, 1998.Google Scholar
  25. 25.
    R. Hartshorne, Algebraic Geometry, Springer, 1977.Google Scholar
  26. 26.
    J. Hietarinta and C. Viallet, Singularity confinement and chaos in discrete systems, Phys. Rev. Lett., 81 (1997), 325–328.CrossRefGoogle Scholar
  27. 27.
    A. Hirschowitz, Symétries des surfaces rationnelles génériques, Math. Ann., 281 (1988), 255–261.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990.Google Scholar
  29. 29.
    S. Kantor, Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Mayer & Müller, 1895.Google Scholar
  30. 30.
    E. Looijenga, Rational surfaces with an anticanonical cycle, Ann. Math., 114 (1981), 267–322.CrossRefMathSciNetGoogle Scholar
  31. 31.
    Y. Manin, Rational surfaces over perfect fields, II, Math. USSR Sb., 1 (1967), 141–168.zbMATHCrossRefGoogle Scholar
  32. 32.
    Y. Manin, Cubic Forms. Algebra, Geometry, Arithmetic, North-Holland Publishing Co., 1986.Google Scholar
  33. 33.
    J. F. McKee, P. Rowlinson, and C. J. Smyth, Pisot numbers from stars, in Number Theory in Progress, vol. I, pp. 309–319, de Gruyter, 1999.Google Scholar
  34. 34.
    C. McMullen, Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math., Inst. Hautes Étud. Sci., 95 (2002), 151–183.zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    C. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math. 545 (2002), 201–233.Google Scholar
  36. 36.
    M. Nagata, On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1, Mem. Coll. Sci. Univ. Kyoto Ser. A Math., 32 (1960), 351–370.zbMATHMathSciNetGoogle Scholar
  37. 37.
    M. Nagata, On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33 (1961), 271–293.Google Scholar
  38. 38.
    V. V. Nikulin, Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, in Proceedings of the International Congress of Mathematicians, pp. 654–671, Amer. Math. Soc., 1986.Google Scholar
  39. 39.
    L. Paris, Irreducible Coxeter groups, Preprint, 2005.Google Scholar
  40. 40.
    R. Penrose and C. A. B. Smith, A quadratic mapping with invariant cubic curve, Math. Proc. Camb. Philos. Soc., 89 (1981), 89–105.zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    R. Rumely, On Bilu’s equidstribution theorem, Contemp. Math., 237 (1999), 159–166.MathSciNetGoogle Scholar
  42. 42.
    R. Salem, Algebraic Numbers and Fourier Analysis, Wadsworth, 1983.Google Scholar
  43. 43.
    J. Silverman, Rational points on K3 surfaces: A new canonical height, Invent. math., 105 (1991), 347–373.zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    T. Takenawa, Algebraic entropy and the space of initial values for discrete dynamical systems. J. Phys. A, 34 (2001), 10533–10545.Google Scholar
  45. 45.
    P. Du Val, On the Kantor group of a set of points in a plane, Proc. London Math. Soc., 42 (1936), 18–51.zbMATHCrossRefGoogle Scholar
  46. 46.
    L. Wang, Rational points and canonical heights on K3-surfaces in P 1×P 1×P 1, in Recent Developments in the Inverse Galois Problem, pp. 273–289, Amer. Math. Soc., 1995.Google Scholar
  47. 47.
    D.-Q. Zhang, Automorphisms of finite order on rational surfaces. With an appendix by I. Dolgachev, J. Algebra 238 (2001), 560–589.Google Scholar

Copyright information

© IHES and Springer-Verlag 2007

Authors and Affiliations

  • Curtis T. McMullen
    • 1
  1. 1.Mathematics DepartmentHarvard UniversityCambridgeUSA

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