Proceedings of the American Mathematical Society

Author(s): Cho-Ho Chu; Pauline Mellon
Journal: Proc. Amer. Math. Soc. 125 (1997), 1771-1777.
MSC (1991): Primary 46G20, 32A10, 32A17; Secondary 32M15
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Abstract: Given a compact holomorphic fixed-point-free self-map, $f$

, of the open unit ball of a Hilbert space, we show that the sequence of iterates, $(f^n)$

, converges locally uniformly to a constant map $\xi$ with $\Vert \xi \Vert = 1$ . This extends results of Denjoy (1926), Wolff (1926), Hervé (1963) and MacCluer (1983). The result is false without the compactness assumption, nor is it true for all open balls of $J^{*}$ -algebras.

1.: M. Abate, Iteration theory, compactly divergent sequences and commuting holomorphic maps, Ann. della Scuola Normale Superiore di Pisa 18 (1991) 67-191. MR 93e:32034a
2.: R. B. Burckel, Iterating analytic self-maps of discs, Amer. Math. Monthly 88 (1981) 396-407. MR 82g:30046
3.: A. Denjoy, Sur l'itération des fonctions analytiques, C. R. Acad. Sc. Paris 182 (1926) 255-257.
4.: S. Dineen, ``The Schwarz lemma'', Oxford Univ. Press, Oxford, 1989. MR 91f:46064
5.: C. J. Earle and R. S. Hamilton, A fixed point theorem for holomorphic mappings, Proc. Symposia Pure Math 16 (1969) 61-65. MR 42:918
6.: M. P. Fatou, Sur l'équations fonctionnelles, Bull. Soc. Math. France 47 (1919) 161-271.
7.: M. P. Fatou, Sur l'équations fonctionnelles, Bull. Soc. Math. France 48 (1920) 22-94 and 208-314.
8.: K. Goebel, Fixed points and invariant domains of holomorphic mappings of the Hilbert ball, Nonlinear Analysis 6 (1982) 1327-1334. MR 84j:58027
9.: K. Goebel and S. Reich, ``Uniform convexity, hyperbolic geometry and nonexpansive mappings'', Marcel Dekker, 1984. MR 86d:58012
10.: K. Goebel and S. Reich, Iterating holomorphic self-mappings of the Hilbert ball, Proc. Japan. Acad. 58 (1982) 349-352. MR 84b:47065
11.: L. A. Harris, A generalization of $C^{*}$ -algebras, Proc. London Math. Soc. 42 (1981) 331-361. MR 82e:46089
12.: T. L. Hayden and T. J. Suffridge, Biholomorphic maps in Hilbert space have a fixed point, Pacific J. Math. 38 (1971) 419-422. MR 46:4288
13.: M. Hervé, Quelques propriétés des applications analytiques d'une boule à dimensions dans elle-meme, J. Math. Pures et Appl. 42 (1963) 117-147. MR 28:3177
14.: G. Julia, Mémoire sur l'itération des fonctions rationnelles, J. Math. Pures et Appl. 8 (1918) 47-245.
15.: B. D. MacCluer, Iterates of holomorphic self-maps of the open unit ball in $\mathbb {C}^n$ , Michigan Math J. 30 (1983) 97-106. MR 85c:32047a
16.: P. Mellon, Another look at results of Wolff and Julia type for $J^{*}$ -algebras, J. Math. Anal. Appl. 198 (1996) 444-457. CMP 96:08
17.: W. Rudin, ``Function theory in the unit ball of $\mathbb {C}^n\>$ '', Springer, New York, 1980. MR 82i:32002
18.: A. Stachura, Iterates of holomorphic self-maps of the unit ball in Hilbert space, Proc. Amer. Math. Soc. 93 (1985) 88-90. MR 86b:47117
19.: H. Upmeier, ``Symmetric Banach manifolds and Jordan $C^{*}$ -algebras'', North-Holland, 1985. MR 87a:58022
20.: E. Vesentini, Iteration of holomorphic maps, Russ. Math. Survey 40 (1985) 7-11.
21.: J. P. Vigué, Sur le groupe des automorphismes analytiques d'un ouvert borné d'un espace de Banach complexe, C. R. Acad. Sc. Paris 278 (1974) 617-620. MR 49:3230
22.: J. P. Vigué, Le groupe des automorphismes analytiques d'un domaine borné d'un espace de Banach complexe, application aux domaines bornés symétriques, Ann. Sc. Ec. Norm. Sup. 9 (1976) 203-282. MR 55:3340
23.: J. Wolff, Sur l'itération des fonctions bornées, C. R. Acad. Sc. Paris 182 (1926) 42-43 and 200-201.
24.: J. Wolff, Sur une généralisation d'un théorème de Schwarz, C. R. Acad. Sc. Paris 182 (1926) 918-920 and 183 (1926) 500-502.
25.: K. W{\l}odarczyk, Julia's lemma and Wolff's theorem for $J^{*}$ -algebras, Proc. Amer. Math. Soc. 99 (1987) 472-476. MR 88a:46049

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