Iteration of compact holomorphic maps on a Hilbert ball

Chu, Cho-Ho; Mellon, Pauline

doi:10.1090/S0002-9939-97-03761-1

Iteration of compact holomorphic maps on a Hilbert ball
HTML articles powered by AMS MathViewer

by Cho-Ho Chu and Pauline Mellon PDF

Proc. Amer. Math. Soc. 125 (1997), 1771-1777 Request permission

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.

References

Marco Abate, Iteration theory, compactly divergent sequences and commuting holomorphic maps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), no. 2, 167–191. MR 1129300
R. B. Burckel, Iterating analytic self-maps of discs, Amer. Math. Monthly 88 (1981), no. 6, 396–407. MR 622955, DOI 10.2307/2321822
A. Denjoy, Sur l’itération des fonctions analytiques, C. R. Acad. Sc. Paris 182 (1926) 255-257.
Seán Dineen, The Schwarz lemma, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Oxford Science Publications. MR 1033739
Clifford J. Earle and Richard S. Hamilton, A fixed point theorem for holomorphic mappings, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 61–65. MR 0266009
M. P. Fatou, Sur l’équations fonctionnelles, Bull. Soc. Math. France 47 (1919) 161-271.
M. P. Fatou, Sur l’équations fonctionnelles, Bull. Soc. Math. France 48 (1920) 22-94 and 208-314.
K. Goebel, Fixed points and invariant domains of holomorphic mappings of the Hilbert ball, Nonlinear Anal. 6 (1982), no. 12, 1327–1334. MR 684968, DOI 10.1016/0362-546X(82)90107-9
Kazimierz Goebel and Simeon Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. MR 744194
Kazimierz Goebel and Simeon Reich, Iterating holomorphic self-mappings of the Hilbert ball, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 8, 349–352. MR 683261
Lawrence A. Harris, A generalization of $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 42 (1981), no. 2, 331–361. MR 607306, DOI 10.1112/plms/s3-42.2.331
T. L. Hayden and T. J. Suffridge, Biholomorphic maps in Hilbert space have a fixed point, Pacific J. Math. 38 (1971), 419–422. MR 305158, DOI 10.2140/pjm.1971.38.419
Michel Hervé, Quelques propriétés des applications analytiques d’une boule à $m$ dimensions dan elle-même, J. Math. Pures Appl. (9) 42 (1963), 117–147 (French). MR 159962
G. Julia, Mémoire sur l’itération des fonctions rationnelles, J. Math. Pures et Appl. 8 (1918) 47-245.
Barbara D. MacCluer, Iterates of holomorphic self-maps of the unit ball in $\textbf {C}^{N}$, Michigan Math. J. 30 (1983), no. 1, 97–106. MR 694933, DOI 10.1307/mmj/1029002792
P. Mellon, Another look at results of Wolff and Julia type for $J\star$-algebras, J. Math. Anal. Appl. 198 (1996) 444–457.
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594, DOI 10.1007/978-1-4613-8098-6
Adam Stachura, Iterates of holomorphic self-maps of the unit ball in Hilbert space, Proc. Amer. Math. Soc. 93 (1985), no. 1, 88–90. MR 766533, DOI 10.1090/S0002-9939-1985-0766533-5
Harald Upmeier, Symmetric Banach manifolds and Jordan $C^\ast$-algebras, North-Holland Mathematics Studies, vol. 104, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 96. MR 776786
E. Vesentini, Iteration of holomorphic maps, Russ. Math. Survey 40 (1985) 7-11.
Jean-Pierre Vigué, Sur le groupe des automorphismes analytiques d’un ouvert borné d’un espace de Banach complexe, C. R. Acad. Sci. Paris Sér. A 278 (1974), 617–620 (French). MR 338466
Jean-Pierre Vigué, Le groupe des automorphismes analytiques d’un domaine borné d’un espace de Banach complexe. Application aux domaines bornés symétriques, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 2, 203–281. MR 430335, DOI 10.24033/asens.1310
J. Wolff, Sur l’itération des fonctions bornées, C. R. Acad. Sc. Paris 182 (1926) 42-43 and 200-201.
J. Wolff, Sur une généralisation d’un théorème de Schwarz, C. R. Acad. Sc. Paris 182 (1926) 918-920 and \text183 (1926) 500-502.
Kazimierz Włodarczyk, Julia’s lemma and Wolff’s theorem for $J^\ast$-algebras, Proc. Amer. Math. Soc. 99 (1987), no. 3, 472–476. MR 875383, DOI 10.1090/S0002-9939-1987-0875383-2