Common fixed points in hyperbolic Riemann surfaces and convex domains

Abate, Marco; Vigué, Jean-Pierre

doi:10.1090/S0002-9939-1991-1065938-8

Common fixed points in hyperbolic Riemann surfaces and convex domains
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by Marco Abate and Jean-Pierre Vigué

Proc. Amer. Math. Soc. 112 (1991), 503-512

DOI: https://doi.org/10.1090/S0002-9939-1991-1065938-8

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Abstract:

In this paper we prove that a commuting family of continuous self-maps of a bounded convex domain in ${\mathbb {C}^n}$ which are holomorphic in the interior has a common fixed point. The proof makes use of three basic ingredients: iteration theory of holomorphic maps, a precise description of the structure of the boundary of a convex domain, and a similar result for commuting families of self-maps of a hyperbolic domain of a compact Riemann surface.

References

Marco Abate, Horospheres and iterates of holomorphic maps, Math. Z. 198 (1988), no. 2, 225–238. MR 939538, DOI 10.1007/BF01163293
Marco Abate, Iteration theory on weakly convex domains, Seminars in Complex Analysis and Geometry, 1988 (Arcavacata, 1988) Sem. Conf., vol. 4, EditEl, Rende, 1990, pp. 1–16. MR 1222246
Marco Abate, Common fixed points of commuting holomorphic maps, Math. Ann. 283 (1989), no. 4, 645–655. MR 990593, DOI 10.1007/BF01442858
Marco Abate, Iteration theory of holomorphic maps on taut manifolds, Research and Lecture Notes in Mathematics. Complex Analysis and Geometry, Mediterranean Press, Rende, 1989. MR 1098711
William M. Boyce, Commuting functions with no common fixed point, Trans. Amer. Math. Soc. 137 (1969), 77–92. MR 236331, DOI 10.1090/S0002-9947-1969-0236331-5
Henri Cartan, Sur les rétractions d’une variété, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 14, 715 (French, with English summary). MR 870703
Carl C. Cowen, Commuting analytic functions, Trans. Amer. Math. Soc. 283 (1984), no. 2, 685–695. MR 737892, DOI 10.1090/S0002-9947-1984-0737892-8
Dan J. Eustice, Holomorphic idempotents and common fixed points on the $2$-disk, Michigan Math. J. 19 (1972), 347–352. MR 313486
L. F. Heath and T. J. Suffridge, Holomorphic retracts in complex $n$-space, Illinois J. Math. 25 (1981), no. 1, 125–135. MR 602903
John Philip Huneke, On common fixed points of commuting continuous functions on an interval, Trans. Amer. Math. Soc. 139 (1969), 371–381. MR 237724, DOI 10.1090/S0002-9947-1969-0237724-2
Peter Kiernan, Hyperbolically imbedded spaces and the big Picard theorem, Math. Ann. 204 (1973), 203–209. MR 372260, DOI 10.1007/BF01351589
Tadeusz Kuczumow and Adam Stachura, Iterates of holomorphic and $k_D$-nonexpansive mappings in convex domains in $\textbf {C}^n$, Adv. Math. 81 (1990), no. 1, 90–98. MR 1051224, DOI 10.1016/0001-8708(90)90005-8
Raghavan Narasimhan, Several complex variables, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1971. MR 0342725
Hugo Rossi, Vector fields on analytic spaces, Ann. of Math. (2) 78 (1963), 455–467. MR 162973, DOI 10.2307/1970536
Allen L. Shields, On fixed points of commuting analytic functions, Proc. Amer. Math. Soc. 15 (1964), 703–706. MR 165508, DOI 10.1090/S0002-9939-1964-0165508-3
T. J. Suffridge, Common fixed points of commuting holomorphic maps of the hyperball, Michigan Math. J. 21 (1974), 309–314. MR 367661
Jean-Pierre Vigué, Géodésiques complexes et points fixes d’applications holomorphes, Adv. in Math. 52 (1984), no. 3, 241–247 (French). MR 744858, DOI 10.1016/0001-8708(84)90023-9
Jean-Pierre Vigué, Points fixes d’applications holomorphes dans un domaine borné convexe de $\textbf {C}^n$, Trans. Amer. Math. Soc. 289 (1985), no. 1, 345–353 (French, with English summary). MR 779068, DOI 10.1090/S0002-9947-1985-0779068-5