Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Complex geodesics and iterates of holomorphic maps on convex domains in $ {\bf C}\sp n$


Author: Peter R. Mercer
Journal: Trans. Amer. Math. Soc. 338 (1993), 201-211
MSC: Primary 32H50; Secondary 32H15, 32H40
DOI: https://doi.org/10.1090/S0002-9947-1993-1123457-0
MathSciNet review: 1123457
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study complex geodesics $ f:\Delta \to \Omega $, where $ \Delta $ is the unit disk in $ {\mathbf{C}}$ and $ \Omega $ belongs to a class of bounded convex domains in $ {{\mathbf{C}}^n}$ with no boundary regularity assumption. Along with continuity up to the boundary, existence of such complex geodesics with two prescribed values $ z$, $ w \in \bar \Omega $ is established. As a consequence we obtain some new results from iteration theory of holomorphic self maps of bounded convex domains in $ {{\mathbf{C}}^n}$.


References [Enhancements On Off] (What's this?)

  • [A1] M. Abate, Boundary behavior of invariant distances and complex geodesics, Rend. Accad. Naz. Lincei 80 (1986), 100-106. MR 976695 (90c:32034)
  • [A2] -, Horospheres and iterates of holomorphic maps, Math. Z. 198 (1988), 225-238. MR 939538 (90e:32028)
  • [A3] -, Common fixed points of commuting holomorphic maps, Math. Ann. 283 (1989), 645-655. MR 990593 (90k:32074)
  • [A4] -, The Lindelöf principle and the angular derivative in strongly convex domains, J. Analyse Math. 54 (1990), 189-228. MR 1041181 (91d:32011)
  • [A5] -, Iteration theory of holomorphic maps on taut manifolds, Mediterranean Press, Rende, Cosenza, 1989. MR 1098711 (92i:32032)
  • [A6] -, Iteration theory, compactly divergent sequences and commuting holomorphic maps, Preprint, 1990.
  • [B] R. B. Burckel, Iterating self-maps of the disk, Amer. Math. Monthly 88 (1981), 396-407. MR 622955 (82g:30046)
  • [CHL] C. H. Chang, M. C. Hu, and H. P. Lee, Extremal analytic discs with prescribed boundary data, Trans. Amer. Math. Soc. 310 (1988), 355-369. MR 930081 (89f:32043)
  • [C] G. N. Chen, Iteration of holomorphic maps of the open unit ball and the generalized upper half plane of $ {{\mathbf{C}}^n}$, J. Math. Anal. Appl. 98 (1984), 305-313. MR 730507 (85e:32001)
  • [D] A. Denjoy, Sur l'itération des fonctions analytiques, C. R. Acad. Sci. Paris 182 (1926), 255-257.
  • [DT] S. Dineen and R. M. Timoney, Complex geodesics on convex domains, Preprint, 1990. MR 1150757 (92m:46066)
  • [GP] F. W. Gehring and B. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172-199. MR 0437753 (55:10676)
  • [GO] G. M. Goluzin, Geometric theory of functions of a complex variable, Transl. Math. Monos., vol. 26, Amer. Math. Soc., Providence, R.I., 1969. MR 0247039 (40:308)
  • [G] I. Graham, Distortion theorems for holomorphic maps between convex domains in $ {{\mathbf{C}}^n}$, Complex Variables Theory Appl. 15 (1990), 37-42. MR 1055937 (91e:32018)
  • [K1] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Dekker, New York, 1970. MR 0277770 (43:3503)
  • [K2] -, Intrinsic distances, measures and geometric function theory. Bull. Amer. Math. Soc. 82 (1976), 357-416. MR 0414940 (54:3032)
  • [KR] S. G. Krantz, Function theory of several complex variables, Wiley, New York, 1982. MR 635928 (84c:32001)
  • [KU] Y. Kubota, Iteration of holomorphic maps of the unit ball into itself, Proc. Amer. Math. Soc. 88 (1983), 476-480. MR 699417 (85c:32047b)
  • [L1] L. Lempert, La métrique de Kobayashi et la réprésentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474. MR 660145 (84d:32036)
  • [L2] -, Intrinsic distances and holomorphic retracts, Complex Analysis and Applications, 1981, Bulgar. Acad. Sci., Sofia, 1984, pp. 341-364. MR 883254 (87m:32051)
  • [M] B. D. MacCluer, Iterates of holomorphic self-maps of the unit ball in $ {{\mathbf{C}}^n}$, Michigan Math. J. 30 (1983), 97-106. MR 694933 (85c:32047a)
  • [RW] H. Royden and P. M. Wong, Carathéodory and Kobayashi metric on convex domains, Preprint, 1983.
  • [TW] E. Thorp and R. Whitley, The strong maximum modulus theorem for analytic functions into a Banach space, Proc. Amer. Math. Soc. 18 (1987), 640-646. MR 0214794 (35:5643)
  • [VE] W. A. Veech, A second course in complex analysis, Benjamin, New York and Amsterdam, 1967. MR 0220903 (36:3955)
  • [V] E. Vesentini, Complex geodesics, Compositio Math. 44 (1981), 375-394. MR 662466 (84a:32037)
  • [W1] J. Wolff, Sur une généralization d'un théorème de Schwarz, C. R. Acad. Sci. Paris 182 (1926), 918-920.
  • [W2] -, Sur l'itération des fonctions bornées, C. R. Acad. Sci. Paris 1982 (1926), 200-201.
  • [Y] P. Yang, Holomorphic curves and boundary regularity of biholomorphic maps, Preprint, 1978.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32H50, 32H15, 32H40

Retrieve articles in all journals with MSC: 32H50, 32H15, 32H40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1123457-0
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society