Skip to main content

Invertible Dynamics on Blow-ups of \({\mathbb {P}}^{k}\)

  • Conference paper
  • First Online:
Complex Analysis and Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 144))

Abstract

This is a survey of some recent results on the iteration of (pseudo) automorphisms of blowups of k-dimensional projective space.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Amerik, E.: A computation of invariants of a rational self-map. Ann. Fac. Sci. Toulouse Math. (6) 18(3), 445–457 (2009)

    Google Scholar 

  2. Bayraktar, T., Cantat, S.: Constraints on automorphism groups of higher dimensional manifolds. J. Math. Anal. Appl. 405(1), 209–213 (2013)

    Article  MathSciNet  Google Scholar 

  3. Bedford, E.: The dynamical degrees of a mapping. In: Proceedings of the Workshop Future Directions in Difference Equations, pp. 3–13, Colecc. Congr., vol. 69, Univ. Vigo, Serv. Publ., Vigo (2011)

    Google Scholar 

  4. Bedford, E., Cantat, S., Kim, K.: Pseudo-automorphisms with no invariant foliation. J. Mod. Dyn. 8, 221–250 (2014)

    Article  MathSciNet  Google Scholar 

  5. Bedford, E., Diller, J.: Real and complex dynamics of a family of birational maps of the plane: the golden mean subshift. Am. J. Math. 127(3), 595–646 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bedford, E., Diller, J., Kim, K.: Pseudoautomorphisms with invariant curves. In: Proceedings of the Abel Symposium 2013, to appear. arXiv:1401.2386 (2013)

  7. Bedford, E., Kim, K.: On the degree growth of birational mappings in higher dimension. J. Geom. Anal. 14(4), 567–596 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bedford, E., Kim, K.: Periodicities in linear fractional recurrences: degree growth of birational surface maps. Mich. Math. J. 54(3), 647–670 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bedford, E., Kim, K.: Dynamics of rational surface automorphisms: linear fractional recurrences. J. Geom. Anal. 19(3), 553–583 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bedford, E., Kim, K.: Continuous families of rational surface automorphisms with positive entropy. Math. Ann. 348(3), 667–688 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bedford, E., Kim, K.: Dynamics of (pseudo) automorphisms of 3-space: periodicity versus positive entropy. Publ. Mat. 58(1), 65–119 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Blanc, J., Cantat, S.: Dynamical degrees of birational transformations of projective surfaces. arXiv:1307.0361

  13. Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat. 6(1965), 103–144 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cantat, S.: Dynamique des automorphismes des surfaces projectives complexes. C. R. Acad. Sci. Paris Sér. I Math. 328(10), 901–906 (1999)

    Google Scholar 

  15. Cantat, S.: Quelques aspects des systèmes dynamiques polynomiaux: existence, exemples, rigidité. pp. 13–95, Panor. Synthèses, vol. 30, Soc. Math. France, Paris (2010)

    Google Scholar 

  16. Csörnyei, M., Laczkovich, M.: Some periodic and non-periodic recursions. Monatsh. Math. 132(3), 215–236 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. de Fernex, T., Ein, L.: Resolution of indeterminacy of pairs. In: Algebraic Geometry, pp. 165–177. de Gruyter, Berlin (2002)

    Google Scholar 

  18. Diller, J.: Cremona transformations, surface automorphisms, and plane cubics (With an appendix by Dolgachev, I.). Mich. Math. J. 60(2), 409–440 (2011)

    Google Scholar 

  19. Diller, J., Favre, C.: Dynamics of bimeromorphic maps of surfaces. Am. J. Math. 123(6), 1135–1169 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dinh, T.-C., Nguyên, V.-A.: Comparison of dynamical degrees for semi-conjugate meromorphic maps. Comment. Math. Helv. 86(4), 817–840 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dinh, T-C., Nguyên, V.-A., Truong, T.T.: On the dynamical degrees of meromorphic maps preserving a fibration. Commun. Contemp. Math. 14(6), 1250042, 18 pp (2012)

    Google Scholar 

  22. Dinh, T.-C., Sibony, N.: Green currents for holomorphic automorphisms of compact Kähler manifolds. J. Am. Math. Soc. 18(2), 291–312 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dinh, T.-C., Sibony, N.: Une borne supérieure pour l’entropie topologique d’une application rationnelle. Ann. Math. (2) 161(3), 1637–1644 (2005)

    Google Scholar 

  24. Dolgachev, I.: Reflection groups in algebraic geometry. Bull. Am. Math. Soc. (N.S.) 45(1), 1–60 (2008)

    Google Scholar 

  25. Dolgachev, I.: Infinite Coxeter groups and automorphisms of algebraic surfaces. In: The Lefschetz centennial conference, Part I (Mexico City, 1984), pp. 91–106, Contemp. Math., vol. 58, Am. Math. Soc., Providence, RI (1986)

    Google Scholar 

  26. Dolgachev, I.: Weyl groups and Cremona transformations. In: Singularities, Part 1 (Arcata, Calif., 1981), pp. 283–294, Proc. Sympos. Pure Math., vol. 40, Am. Math. Soc., Providence, RI (1983)

    Google Scholar 

  27. Dujardin, R.: Structure properties of laminar currents on \({ P}^{2}\). J. Geom. Anal. 15(1), 25–47 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dujardin, R.: Sur l’intersection des courants laminaires. Publ. Mat. 48(1), 107–125 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dujardin, R.: Laminar currents in \({ P}^{2}\). Math. Ann. 325(4), 745–765 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Favre, C., Wulcan, E.: Degree growth of monomial maps and McMullen’s polytope algebra. Indiana Univ. Math. J. 61(2), 493–524 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Fornæss, J.E., Sibony, N.: Complex dynamics in higher dimensions. Notes partially written by Gavosto. E.A.: NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, Complex potential theory (Montreal, PQ, pp. 131–186. Kluwer Acad. Publ, Dordrecht (1993) (1994)

    Google Scholar 

  32. Fornæss, J.E., Sibony, N.: Complex dynamics in higher dimension. II. In: Modern methods in complex analysis (Princeton, NJ, 1992), pp. 135–182, Ann. of Math. Stud., vol. 137, Princeton Univ. Press, Princeton, NJ (1995)

    Google Scholar 

  33. Freire, A., Lopes, A., Mañé, R.: An invariant measure for rational maps. Bol. Soc. Brasil. Mat. 14(1), 45–62 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Guedj, V.: Entropie topologique des applications méromorphes. Ergodic Theory Dyn. Syst. 25(6), 1847–1855 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Guedj, V.: Propriétés ergodiques des applications rationelles. In: Quelques aspects des systèmes dynamiques polynômiaux, pp. 97–202, Panor. Synthèses, vol. 30, Soc. Math. France, Paris (2010)

    Google Scholar 

  36. Kaschner, S.R., Pérez, R.A., Roeder, R.K.W.: Examples of rational maps of \({\mathbb{C}}P^{2}\) with equal dynamical degrees and no invariant foliation, arXiv:1309.4364

  37. Koch, S., Roeder, R.: Computing dynamical degrees, arXiv:1403.5840

  38. Lin, J.-L.: Pulling back cohomology classes and dynamical degrees of monomial maps. Bull. Soc. Math. France 140(4), 533–549 (2013) 2012

    Google Scholar 

  39. Lyness, R.C.: Notes 1581,1847, and 2952, Math. Gazette 26 (1942), 62, 29 (1945), 231, and 45 (1961), 201

    Google Scholar 

  40. Lyubich, M.: Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dyn. Syst. 3(3), 351–385 (1983)

    MathSciNet  MATH  Google Scholar 

  41. McMullen, C.T.: Dynamics on blowups of the projective plane. Publ. Math. Inst. Hautes tudes Sci. No. 105, pp. 49–89 (2007)

    Google Scholar 

  42. Mukai, S.: Geometric realization of \(T\)-shaped root systems and counterexamples to Hilbert’s tenth problem. In: Algebraic transformation groups and algebraic varieties, vol. 132 of Encyclopedia Math. Sci., pp. 123–129. Springer, Berlin (2004)

    Google Scholar 

  43. Oguiso, K., Truong, T.T.: Salem numbers in dynamics of Kähler threefolds and complex tori, arXiv:1309.4851

  44. Oguiso, K., Truong, T.T.: Explicit Examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy, arXiv:1306.1590

  45. 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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Truong, T.T.: On automorphisms of blowups of \({\bf P}^{3}\), arXiv:1202.4224

  47. Uehara, T.: Rational surface automorphisms with positive entropy, arXiv:1009.2143

  48. Xie, J.: Periodic points of birational maps on projective surfaces, arXiv:1106.1825

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Eric Bedford .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Japan

About this paper

Cite this paper

Bedford, E. (2015). Invertible Dynamics on Blow-ups of \({\mathbb {P}}^{k}\) . In: Bracci, F., Byun, J., Gaussier, H., Hirachi, K., Kim, KT., Shcherbina, N. (eds) Complex Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol 144. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55744-9_5

Download citation

Publish with us

Policies and ethics