Skip to main content
SpringerLink
Log in

Dynamique des applications holomorphes propres de domaines réguliers et problème de l'injectivité

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Summary

This paper deals with proper holomorphic self-maps of smoothly bounded pseudoconvex domains in . We study the dynamical properties of their extension to the boundary and show that their non-wandering sets are always contained in the weakly pseudoconvex part of the boundary. In the case of complete circular domains, we combine this fact with an entropy/degree argument to show that the maps are automorphisms. Some of our results remain true in

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abate, M.: Iteration theory of holomorphic maps on taut manifolds. Research and Lecture Notes in Mathematics. Complex Analysis and Geometry, Mediterranean Press, Rende, 1989

  2. Alexander, H.: Proper holomorphic mappings in C n. Indiana Univ. Math. J. 26(1), 137–146 (1977)

    Article  MathSciNet  Google Scholar 

  3. Bedford, E., Bell, S.: Proper self-maps of weakly pseudoconvex domains. Math. Ann. 261(1), 47–49 (1982)

    Article  MathSciNet  Google Scholar 

  4. Bell, S.: Local boundary behavior of proper holomorphic mappings. In: Complex analysis of several variables (Madison, Wis., 1982) Proc. Sympos. Pure Math. vol. 41. Amer. Math. Soc., Providence, RI, 1984, pp 1–7

  5. ADFASDFASDF

  6. Berger, M.: Géométrie. 5. CEDIC, Paris, 1977. La sphère pour elle-mOme, géométrie hyperbolique, l'espace des sphères

  7. Berhanu, S., Mendoza, G.A.: Orbits and global unique continuation for systems of vector fields. J. Geom. Anal. 7(2), 173–194 (1997)

    MathSciNet  Google Scholar 

  8. Berteloot, F.: Holomorphic vector fields and proper holomorphic self-maps of Reinhardt domains. Ark. Mat. 36(2), 241–254 (1998)

    Article  MathSciNet  Google Scholar 

  9. Berteloot, F.,Pinchuk, S.: Proper holomorphic mappings between bounded complete Reinhardt domains in C 2. Math. Z., 219(3), 343–356 (1995)

    MathSciNet  Google Scholar 

  10. Boas, H., Straube, E.: Sobolev estimates for the -Neumann operator on domains in C n admitting a defining function that is plurisubharmonic on the boundary. Math. Z., 206(1), 81–88 (1991)

    MathSciNet  Google Scholar 

  11. Chabat, B.: Introduction à l'analyse complexe. Tome 2, Traduit du Russe: Mathématiques. ``Mir'', Moscow, 1990

  12. Coupet, B., Pan, Y., Sukhov, A.: Proper holomorphic self-maps of quasi-circular domains in C 2. Nagoya Math. J. 164, 1–16 (2001)

    MATH  MathSciNet  Google Scholar 

  13. Diederich, K., Fornæss, J.E.: Proper holomorphic images of strictly pseudoconvex domains. Math. Ann. 259(2), 279–286 (1982)

    Article  MathSciNet  Google Scholar 

  14. Dupont, C.: Exemples de Lattès et domaines faiblement sphériques de Manuscripta Math. 111(3), 357–378 (2003)

  15. Fornæss, J.E., Sibony, N.: Construction of P.S.H. functions on weakly pseudoconvex domains. Duke Math. J. 58(3), 633–655 (1989)

    Google Scholar 

  16. Forstnerič, F.: Proper holomorphic mappings: a survey. Several complex variables (Stockholm, 1987/1988), Math. Notes, vol. 38. Princeton Univ. Press Princeton, NJ, 1993, pp 297–363

  17. Gromov, M.: On the entropy of holomorphic maps. Enseign. Math. (2), 49(3–4), 217–235 (2003)

    Google Scholar 

  18. Henkin, G., Novikov, R.: Proper mappings of classical domains, Vol. 1043. Springer, Berlin, 1984, pp 625–627

  19. Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems, Vol. 54 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1995

  20. Misiurewicz, M., Przytycki, F.: Topological entropy and degree of smooth mappings. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25(6), 573–574 (1977)

    MathSciNet  Google Scholar 

  21. Nagel, A., Stein, E.M., Wainger, S,: Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155(1–2), 103–147 (1985)

    Google Scholar 

  22. Pinchuk, S.: The scaling method and holomorphic mappings. In: Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989), Vol. 52. Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, pp 151–161

  23. Pinchuk, S.I.: Proper holomorphic maps of strictly pseudoconvex domains. Sibirsk. Mat. Ž. 15, 909–917, 959 (1974)

    MathSciNet  Google Scholar 

  24. Pinčuk, S.I.: Proper holomorphic mappings of strictly pseudoconvex domains, Petits diviseurs en dimension 1. Dokl. Akad. Nauk SSSR 241(1), 30–33 (1978)

    MathSciNet  Google Scholar 

  25. Rudin, W.: Function theory in the unit ball of C n, Vol. 241 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science]. Springer-Verlag, New York, 1980

  26. Rudin, W.: New constructions of functions holomorphic in the unit ball of C n, Vol. 63 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986

  27. Sibony, N.: Une classe de domaines pseudoconvexes. Duke Math. J. 55(2), 299–319 (1987)

    Article  MathSciNet  Google Scholar 

  28. Tsai, I.H.: Rigidity of proper holomorphic maps between symmetric domains. J. Differential Geom. 37(1), 123–160 (1993)

    MathSciNet  Google Scholar 

  29. Tumanov, A.E., Khenkin, G.M.: Local characterization of analytic automorphisms of classical domains. Dokl. Akad. Nauk SSSR 267(4), 796–799 (1982)

    MathSciNet  Google Scholar 

  30. Yoccoz, J.-C.: Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Astérisque, 231, 3–88 (1995)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Emile Picard, Université Paul Sabatier, 118 Route de Narbonne, 31062, Toulouse Cedex 04, France

    Emmanuel Opshtein

Authors
  1. Emmanuel Opshtein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emmanuel Opshtein.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Opshtein, E. Dynamique des applications holomorphes propres de domaines réguliers et problème de l'injectivité. Math. Ann. 335, 1–30 (2006). https://doi.org/10.1007/s00208-005-0689-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-005-0689-4

Search

Navigation