Arkiv för Matematik

  • Ark. Mat.
  • Volume 40, Number 2 (2002), 207-243.

Dynamics of polynomial automorphisms of Ck

Vincent Guedj and Nessim Sibony

Full-text: Open access


We study the dynamics of polynomial automorphisms of Ck. To an algebraically stable automorphism we associate positive closed currents which are invariant under f, considering f as a rational map on Pk. These currents give information on the dynamics and allow us to construct a canonical invariant measure which is shown to be mixing.

Article information

Ark. Mat., Volume 40, Number 2 (2002), 207-243.

Received: 17 April 2001
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2002 © Institut Mittag-Leffler


Guedj, Vincent; Sibony, Nessim. Dynamics of polynomial automorphisms of C k. Ark. Mat. 40 (2002), no. 2, 207--243. doi:10.1007/BF02384535.

Export citation


  • [A] Ahlfors, L. V., Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill, New York, 1973.
  • [BLS] Bedford, E., Lyubich, M. and Smillie, J., Distribution of periodic points of polynomial diffeomorphisms of C2, Invent. Math. 114 (1993), 277–288.
  • [BP] Bedford, E. and Pambuccian, V., Dynamics of shift-like diffeomorphisms of Ck, Conform. Geom. Dyn. 2 (1998), 45–55.
  • [BS2] Bedford, E. and Smillie, J., Polynomial diffeomorphisms of C2 III: Ergodicity, exponents and entropy of the equilibrium measure, Math. Ann. 294 (1992), 395–420.
  • [BS3] Bedford, E. and Smillie, J., Extremal rays in the dynamics of polynomial automorphisms of C2, in Complex Geometric Analysis in Pohang (1997) (Kim, K.-T. and Krantz, S. G., eds.), Contemp. Math. 222, pp. 41–79, Amer. Math. Soc., Providence, R. I., 1999.
  • [CG] Carleson, L. and Gamelin, T. W., Complex Dynamics, Springer-Verlag, Berlin-Heidelberg-New York, 1993.
  • [CF] Coman, D. and Fornæss, J. E., Green’s functions for irregular quadratic polynomial automorphisms of C3, Michigan Math. J. 46 (1999), 419–459.
  • [Fa] Favre, C., Note on pull-back and Lelong number of currents, Bull. Soc. Math. France 127 (1999), 445–458.
  • [FG] Favre, C. and Guedj, V., Dynamique des applications rationnelles des espaces multiprojectifs, Indiana Math. J. 50 (2001), 881–934.
  • [Fe] Federer, H., Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg, 1969.
  • [FS1] Fornæss, J. E. and Sibony, N., Complex dynamics in higher dimension, in Complex Potential Theory (Montreal, Que., 1993) (Gauthier, P.-M., ed.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 439, pp. 131–186, Kluwer, Dordrecht, 1994.
  • [FS2] Fornæss, J. E. and Sibony, N., Complex dynamics in higher dimension II, in Modern Methods in Complex Analysis (Princeton, N. J., 1992) (Bloom, T., Catlin, D., D’Angelo, J. P. and Siu, Y.-T., eds.), Ann. Math. Studies 137, pp. 135–187, Princeton Univ. Press, Princeton, N. J., 1995.
  • [FS3] Fornæss, J. E. and Sibony, N., Oka’s inequality for currents and applications, Math. Ann. 301 (1995), 399–419.
  • [Fr] Friedland, S., Entropy of polynomial and rational maps, Ann. of Math. 133 (1991), 350–368.
  • [FM] Friedland, S. and Milnor, J., Dynamical properties of plane automorphisms, Ergodic Theory Dynam. Systems 9 (1998), 67–99.
  • [HP] Harvey, R. and Polking, J., Extending analytic objects, Comm. Pure Appl. Math. 28 (1975), 701–727.
  • [He] Heins, M., On a notion of convexity connected with a method of Carleman, J. Anal. Math. 7 (1959), 53–77.
  • [Hö] Hörmander, L., The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin-Heidelberg, 1983.
  • [L] Lazarsfeld, R., Some applications of the theory of positive vector bundles, in Complete Intersections (Acireale, 1983), Lecture Notes in Math. 1092, pp. 29–61. Springer-Verlag, Berlin-Heidelberg, 1984.
  • [S] Sibony, S., Dynamique des applications rationnelles de Pk, in Dynamique et géométrie complexes (Lyon, 1997), Panorama et Synthèses 8, pp. ix-x, xi–xii, 97–185, Soc. Math. France, Paris, 1999.
  • [Si] Siu, Y.-T., Analyticity of sets associated to Lelong numbers and extension of closed positive currents, Invent. Math. 27, (1974), 53–156.
  • [T] Taylor, B. A., An estimate for an extremal plurisubharmonic function on Cn, in Séminaire d’Analyse P. Lelong-P. Dolbeault-H. Skoda. Années 1981/1983 (Lelong, P., Dolbeault, P. and Skoda, H., eds.), Lecture Notes in Math. 1028, pp. 318–328, Springer-Verlag, Berlin-Heidelberg, 1983.
  • [W] Walters, P., An Introduction to Ergodic Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1982.