Commuting holomorphic functions and hyperbolic automorphisms

de Fabritiis, Chiara

doi:10.1090/S0002-9939-96-03729-X

Commuting holomorphic functions and hyperbolic automorphisms
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by Chiara de Fabritiis

Proc. Amer. Math. Soc. 124 (1996), 3027-3037

DOI: https://doi.org/10.1090/S0002-9939-96-03729-X

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Abstract:

We give a complete classification of the holomorphic self-maps of the unit ball of $\mathbf {C}^{n}$ into itself which commute with a given hyperbolic automorphism.

References

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
Marco Abate and Jean-Pierre Vigué, Common fixed points in hyperbolic Riemann surfaces and convex domains, Proc. Amer. Math. Soc. 112 (1991), no. 2, 503–512. MR 1065938, DOI 10.1090/S0002-9939-1991-1065938-8
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
C. de Fabritiis, G. Gentili, On holomorphic maps which commute with hyperbolic automorphisms, to appear in Advances in Math., 1995.
Tullio Franzoni and Edoardo Vesentini, Holomorphic maps and invariant distances, Notas de Matemática [Mathematical Notes], vol. 69, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 563329
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
Marek Kuczma, Functional equations in a single variable, Monografie Matematyczne, Tom 46, Państwowe Wydawnictwo Naukowe, Warsaw, 1968. MR 0228862
Marek Kuczma, An introduction to the theory of functional equations and inequalities, Prace Naukowe Uniwersytetu Śląskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 489, Uniwersytet Śląski, Katowice; Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1985. Cauchy’s equation and Jensen’s inequality; With a Polish summary. MR 788497
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
A.L. Shields, On fixed points of commuting analytic functions, Proc. Amer. Math. Soc. 15 (1964), 703–706.