A Wong-Rosay type theorem for proper holomorphic self-maps
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 513-524.

In this short paper, we show that the only proper holomorphic self-maps of bounded domains in k whose iterates approach a strictly pseudoconvex point of the boundary are automorphisms of the euclidean ball. This is a Wong-Rosay type theorem for a sequence of maps whose degrees are a priori unbounded.

Dans cette note, nous prouvons que les seules auto-applications holomorphes propres des domaines bornés de k dont les itérées accumulent un point de stricte-pseudoconvexité du bord sont des automorphismes de la boule. Il s’agit d’un résultat de type Wong-Rosay pour une suite d’applications dont les degrés sont à priori non bornés.

DOI: 10.5802/afst.1254
Opshtein, Emmanuel 1

1 Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000 Strasbourg, France.
@article{AFST_2010_6_19_3-4_513_0,
     author = {Opshtein, Emmanuel},
     title = {A {Wong-Rosay} type theorem for proper holomorphic self-maps},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {513--524},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 19},
     number = {3-4},
     year = {2010},
     doi = {10.5802/afst.1254},
     mrnumber = {2790806},
     zbl = {1214.32006},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1254/}
}
TY  - JOUR
AU  - Opshtein, Emmanuel
TI  - A Wong-Rosay type theorem for proper holomorphic self-maps
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2010
DA  - 2010///
SP  - 513
EP  - 524
VL  - Ser. 6, 19
IS  - 3-4
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://archive.numdam.org/articles/10.5802/afst.1254/
UR  - https://www.ams.org/mathscinet-getitem?mr=2790806
UR  - https://zbmath.org/?q=an%3A1214.32006
UR  - https://doi.org/10.5802/afst.1254
DO  - 10.5802/afst.1254
LA  - en
ID  - AFST_2010_6_19_3-4_513_0
ER  - 
%0 Journal Article
%A Opshtein, Emmanuel
%T A Wong-Rosay type theorem for proper holomorphic self-maps
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2010
%P 513-524
%V Ser. 6, 19
%N 3-4
%I Université Paul Sabatier, Institut de mathématiques
%C Toulouse
%U https://doi.org/10.5802/afst.1254
%R 10.5802/afst.1254
%G en
%F AFST_2010_6_19_3-4_513_0
Opshtein, Emmanuel. A Wong-Rosay type theorem for proper holomorphic self-maps. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 513-524. doi : 10.5802/afst.1254. http://archive.numdam.org/articles/10.5802/afst.1254/

[1] Alexander (H.).— Holomorphic mappings from the ball and polydisc. Math. Ann., 209:249-256 (1974). | MR | Zbl

[2] Bell (S.).— Local boundary behavior of proper holomorphic mappings. In Complex analysis of several variables (Madison, Wis., 1982), volume 41 of Proc. Sympos. Pure Math., p. 1-7. Amer. Math. Soc., Providence, RI (1984). | MR | Zbl

[3] Berteloot (F.).— Attraction des disques analytiques et continuité höldérienne d’applications holomorphes propres. In Topics in complex analysis (Warsaw, 1992), volume 31 of Banach Center Publ., p. 91-98. Polish Acad. Sci., Warsaw (1995). | MR | Zbl

[4] Boggess (A.).— CR manifolds and the tangential Cauchy-Riemann complex. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1991). | MR | Zbl

[5] Fornaess (J. E.).— Biholomorphic mappings between weakly pseudoconvex domains. Pacific J. Math., 74(1):63-65 (1978). | MR | Zbl

[6] MacCluer (B. D.).— Iterates of holomorphic self-maps of the unit ball in C N . Michigan Math. J., 30(1):97-106 (1983). | MR | Zbl

[7] Nagel (A.), Stein (E. M.), and Wainger (S.).— Balls and metrics defined by vector fields. I. Basic properties. Acta Math., 155(1-2):103-147 (1985). | MR | Zbl

[8] Opshtein (E.).— Sphericity and contractibility of strictly pseudoconvex hypersurfaces. Prepublication, arXiv math.CV/0504054 (2005).

[9] Opshtein (E.).— Dynamique des applications holomorphes propres des domaines réguliers et problème de l’injectivité. Math. Ann., 133(1):1-30 (2006). | MR | Zbl

[10] Ourimi (N.).— Some compactness theorems of families of proper holomorphic correspondences. Publ. Mat., 47(1):31-43 (2003). | MR | Zbl

[11] Pinčuk (S. I.).— The analytic continuation of holomorphic mappings. Mat. Sb. (N.S.), 98(140)(3(11)):416-435, 495-496 (1975). | MR | Zbl

[12] Rosay (J.-P.).— Sur une caractérisation de la boule parmi les domaines de C n par son groupe d’automorphismes. Ann. Inst. Fourier (Grenoble), 29(4):ix, p. 91-97 (1979). | Numdam | MR | Zbl

[13] Rudin (W.).— Function theory in the unit ball of C n , volume 241 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science]. Springer-Verlag, New York (1980). | MR | Zbl

[14] Tumanov (A. E.) and Khenkin (G. M.).— Local characterization of holomorphic automorphisms of Siegel domains. Funktsional. Anal. i Prilozhen., 17(4):49-61 (1983). | MR | Zbl

[15] Webster (S. M.).— On the transformation group of a real hypersurface. Trans. Amer. Math. Soc., 231(1):179-190 (1977). | MR | Zbl

[16] Wong (B.).— Characterization of the unit ball in C n by its automorphism group. Invent. Math., 41(3):253-257 (1977). | MR | Zbl

Cited by Sources: