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Pluripotential theory on the support of closed positive currents and applications to dynamics in \(\mathbb {C}^n\)

  • Frédéric ProtinEmail author
Article
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Abstract

We extend certain classical theorems in pluripotential theory to a class of functions defined on the support of a (1, 1)-closed positive current T, analogous to plurisubharmonic functions, called T-plurisubharmonic functions. These functions are defined as limits, on the support of T, of sequences of plurisubharmonic functions decreasing on this support. We study these functions by means of a class of measures, so-called pluri-Jensen measures, which prove to be numerous on the support of (1, 1)-closed positive currents. For any fat compact set, we obtain an expression of its relative Green’s function in terms of pluri-Jensen measures and deduce a characterization of the polynomially convex fat compact sets and of pluripolar sets. These tools are then used to study dynamics of a class of automorphisms of \(\mathbb {C}^n\) which naturally generalize Hénon’s automorphisms. We obtain an equidistribution result for the convergence of pull-back of certain measures toward an ergodic invariant measure with compact support.

Keywords

Pluricomplex Green’s function Closed positive current Pluripolar set Polynomial convexity Complex dynamics Jensen measure 

Mathematics Subject Classification

32U15 32E20 32U35 32U40 32U05 32H50 

Notes

Acknowledgements

We thank Prof. E. Bedford and C. Kiselman for their attentive reading.

References

  1. 1.
    Alexander, H.: Projective capacity. Ann. Math. Stud. 100, 3–27 (1981)MathSciNetGoogle Scholar
  2. 2.
    Bayraktar, T.: Equidistribution towards the Green current in big cohomology classes. Int. J. Math. 24, 1350080 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bedford, E., Smillie, J.: Polynomial diffeomorphisms of \(\mathbb{C}^2\). II. J. Am. Math. Soc. 4, 657–679 (1991)zbMATHGoogle Scholar
  4. 4.
    Demailly, J.P.: Monge–Ampere operators, Lelong numbers, and intersection theory. In: Ancona, V., Silva, A. (eds.) Complex Analysis and Geometry, pp. 115–191. Plenum Press, New York (1993)CrossRefGoogle Scholar
  5. 5.
    Dinh, T.C., Dujardin, R., Sibony, N.: On the dynamics near infinity of some polynomial mappings in \(\mathbb{C}^2\). Math. Ann. 333, 703–739 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dinh, T.C., Lawrence, M.G.: Polynomial hulls and positive currents. Ann. Fac. Sci. Toulouse Math. series 6(12), 317–334 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dinh, T.C., Sibony, N.: Dynamique des applications polynomiales semi-régulières. Ark. Mat. 42, 61–85 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Duval, J., Sibony, N.: Polynomial convexity, rational convexity, and currents. Duke Math. J. 79, 487–513 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Edwards, D.A.: Choquet boundary theory for certain spaces of lower semicontinuous functions. In: Birtel (Ed.) Function Algebras. Scott Foresman and Cie, pp. 300–309 (1966)Google Scholar
  10. 10.
    Fornaess, J.E.: The Julia set of Hénon maps. Math. Ann. 334, 457–464 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Favre, C., Guedj, V.: Dynamique des applications rationnelles des espaces multiprojectifs. Indiana Univ. Math. J. 50, 881–934 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fornaess, E., Sibony, N.: Complex dynamics in higher dimensions. In: Complex Potential Theory, NATO ASI series C, 439, pp. 131–18. Gauthier ed. 6 (1994)Google Scholar
  13. 13.
    Fornaess, E., Sibony, N.: Complex dynamics in Higher dimensions II. In: Modern Methods in Complex Analysis, pp. 135–182. Annals of Mathematics Studies 137,(1995)Google Scholar
  14. 14.
    Guedj, V.: Propriétés ergodiques des applications rationnelles. In: Quelques aspects des systèmes dynamiques polynomiaux. Panor. Synthèses 30, 97–202. Soc. Math. France, Paris (2010)Google Scholar
  15. 15.
    Guedj, V., Sibony, N.: Dynamics of polynomials automorphisms of \(\mathbb{C}^k\). Ark. Mat. 40, 207–243 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Guedj, V.: Courants extrémaux et dynamique complexe. Ann. Sci. École Norm. Sup. 38, 407–426 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gamelin, T.W., Rossi, H.: Jensen measures and algebras of analytic functions. In: Birtel, F.T. (Ed.) Function Algebras, pp. 15–35. Scott Foresman and Cie (1966)Google Scholar
  18. 18.
    Klimek, M.: Pluripotential Theory. Clarendon Press, Oxford (1991)zbMATHGoogle Scholar
  19. 19.
    Lasota, A., Mackey, M.C.: Probabilistic Properties of Deterministic Systems. Cambridge University Press, Cambridge (1985)CrossRefzbMATHGoogle Scholar
  20. 20.
    Poletsky, E.A.: Plurisubharmonic functions as solutions of variational problems. In: Several Complex Variables and Complex Geometry (Santa Cruz, CA, 1989), Part. 1, pp. 163–171. American Mathematical Society, Providence (1991)Google Scholar
  21. 21.
    Poletsky, E.A.: Holomorphic currents. Indiana Univ. Math. J. 42, 85–144 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Protin, F.: Dynamique d’endomorphismes polynomiaux de \(\mathbb{C}^k\), PhD thesis, Université Toulouse III (2010)Google Scholar
  23. 23.
    Protin, F.: Equidistribution vers le courant de Green. Ann. Polon. Math. 115, 201–218 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Protin, F.: ŁS condition for filled Julia sets in \(\mathbb{C}\). Annali di Matematica (2018).  https://doi.org/10.1007/s10231-018-0752-x
  25. 25.
    Ransford, T.J.: Jensen measures. In: Arakelian, N., Gauthier, P.M. (eds.) Approximation, Complex Analysis and Potential Theory, pp. 221–237. Kluwer, Dordrecht (2001)CrossRefGoogle Scholar
  26. 26.
    Rosay, J.P.: Poletsky theory of disks on holomorphic manifolds. Indiana Univ. Math. J. 52, 157–169 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sibony, N.: Pfaff systems, currents and hulls. Math. Z. 285, 1107–1123 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Stout, E.L.: Polynomial Convexity, Progress in Mathematics, 261. Birkaüser, Basel (2007)Google Scholar
  29. 29.
    Stout, E.L.: The Theory of Uniform Algebras. Bogden and Quigley (1971)Google Scholar
  30. 30.
    Walters, P.: An Introduction to Ergodic Theory. Springer, Berlin (1981)zbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.INSA de Toulouse, Institut de Mathématiques de ToulouseToulouseFrance

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