Pseudoautomorphisms with Invariant Curves

  • Eric BedfordEmail author
  • Jeffery Diller
  • Kyounghee Kim
Conference paper
Part of the Abel Symposia book series (ABEL, volume 10)


Inspired by constructions of automorphisms on rational surfaces and a recent paper of Perroni and Zhang (Mathematische Annalen 359(1–2):189–209, 2014), we give a concrete construction of pseudoautomorphisms of higher dimensional rational surfaces that have an invariant cuspidal curve and first dynamical degree larger than one. Taking advantage of the group structure on the smooth points of the curve and elementary projective geometry, we arrive at explicit formulas for these pseudoautomorphisms. Though it is not used in our construction, we further indicate a relationship between aspects of our construction with certain coxeter groups.


Coxeter Group Topological Entropy Rational Surface Positive Entropy Dynamical Degree 
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.


  1. 1.
    Bayraktar, T., Cantat, S.: Constraints on automorphism groups of higher dimensional manifolds. J. Math. Anal. Appl. 405(1), 209–213 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bedford, E., Kim, K.: On the degree growth of birational mappings in higher dimension. J. Geom. Anal. 14, 567–596 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bedford, E., Kim, K.: Periodicities in linear fractional recurrences: degree growth of birational surface maps. Mich. Math. J. 54(3), 647–670 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bedford, E., Kim, K.: Dynamics of rational surface automorphisms: linear fractional recurrences. J. Geom. Anal. 19(3), 553–583 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bedford, E., Kim, K.: Continuous families of rational surface automorphisms with positive entropy. Math. Ann. 348(3), 667–688 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bedford, E., Cantat, S., Kim, K.: Pseudo-automorphisms with no invariant foliation. J. Mod. Dyn. 8(2), 221–250 (2014), ArXiv:1309.3695Google Scholar
  7. 7.
    Cantat, S.: Dynamique des automorphismes des surfaces complexes compactes. These, Ecole Normale Supérieure de Lyon (1999)Google Scholar
  8. 8.
    Cantat, S.: Dynamique des automorphismes des surfaces k3. Acta Math. 187(1), 1–57 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    De Thélin, H., Vigny, G.: Entropy of meromorphic maps and dynamics of birational maps. Mém. Soc. Math. Fr. (N.S.) 122(2010), vi+98 pp.Google Scholar
  10. 10.
    Diller, J.: Cremona transformations, surface automorphisms, and plane cubics. Mich. Math. J. 60(2), 409–440 (2011). With an appendix by Igor Dolgachev.Google Scholar
  11. 11.
    Dolgachev, I.: Reflection groups in algebraic geometry. Bull. Am. Math. Soc. 45(1), 1–60 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Dolgachev, I., Ortland, D.: Point Sets in Projective Spaces and Theta Functions. Société Mathématique de France, Paris (1989)Google Scholar
  13. 13.
    Gromov, M.: On the entropy of holomorphic maps. Enseign. Math. (2) 49(3–4), 217–235 (2003)Google Scholar
  14. 14.
    Guedj, V.: Propriétés ergodiques des applications rationnelles. In: Quelques aspects des systèmes dynamiques polynomiaux. Volume 30 of Panor. Synthèses, pp. 97–202. Société mathématique de France, Paris (2010)Google Scholar
  15. 15.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)Google Scholar
  16. 16.
    McMullen, C.T.: Coxeter groups, Salem numbers and the Hilbert metric. Publ. Math. Inst. Hautes Études Sci. 95, 151–183 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    McMullen, C.T.: Dynamics on k3 surfaces: Salem numbers and siegel disks. J. Reine Angew. Math. 545, 201–233 (2002)zbMATHMathSciNetGoogle Scholar
  18. 18.
    McMullen, C.T.: Dynamics on blowups of the projective plane. Publ. Math. Inst. Hautes Études Sci. 105, 49–90 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Mukai, S.: Geometric realization of T-shaped root systems and counterexamples to Hilbert’s fourteenth problem. In: Algebraic Transformation Groups and Algebraic Varieties. Volume 132 of Encyclopaedia of Mathematical Sciences, pp. 123–129. Springer, Berlin (2004)Google Scholar
  20. 20.
    Nagata, M.: On rational surfaces, ii. Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33(2), 217–293 (1960)Google Scholar
  21. 21.
    Oguiso, K., Truong, T.T.: Salem numbers in dynamics of kähler threefolds and complex tori (2013). ArXiv:1309.4851Google Scholar
  22. 22.
    Perroni, F., Zhang, D.-Q.: Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces. Math. Ann. 359(1–2), 189–209 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Truong, T.T.: On automorphisms of blowups of \(\mathbb{P}^{3}\) (2012). ArXiv:1202.4224Google Scholar
  24. 24.
    Uehara, T.: Rational surface automorphisms with positive entropy (2010). ArXiv:1009.2143Google Scholar
  25. 25.
    Yomdin, Y.: Volume growth and entropy. Isr. J. Math. 57(3):285–300 (1987)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Stony Brook UniversityStony BrookUSA
  3. 3.Department of MathematicsUniversity of Notre DameNotre DameUSA
  4. 4.Department of MathematicsFlorida State UniversityTallahasseeUSA

Personalised recommendations