Abstract
Abstract basins appear naturally in different areas of several complex variables. In this survey we want to describe three different topics in which they play an important role, leading to interesting open problems.
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References
Abate, M., Abbondandolo, A., Majer, P.: Stable manifolds for holomorphic automorphisms. J. Reine Angew. Math. 690, 217–247 (2014)
Abbondandolo, A., Arosio, L., Fornæss, J.E., Majer, P., Peters, H., Raissy, J., Vivas, L.: A survey on non-autonomous basins in several complex variables. arXiv:1311.3835 (preprint)
Abbondandolo, A., Majer, P.: Global stable manifolds in holomorphic dynamics under bunching conditions. Int. Math. Res. Not. IMRN 14, 4001–4048 (2014)
Abate, M., Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: The evolution of Loewner’s differential equations. Newslett. Eur. Math. Soc. 78, 31–38 (2010)
Arosio, L.: Resonances in Loewner equations. Adv. Math. 227, 1413–1435 (2011)
Arosio, L.: Basins of attraction in Loewner equations. Ann. Acad. Sci. Fenn. Math. 37(2), 563–570 (2012)
Arosio, L.: Loewner equations on complete hyperbolic domains. J. Math. Anal. Appl. 398(2), 609–621 (2013)
Arosio, L., Bracci, F.: Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds. Anal. Math. Phys. 1(4), 337–350 (2011)
Arosio, L., Bracci, F.: Canonical models in holomorphic iteration. Trans. Am. Math. Soc. doi:10.1090/tran/6593. arXiv:1401.6873
Arosio, L., Bracci, F., Hamada, H., Kohr, G.: An abstract approach to Loewner chains. J. Anal. Math. 119, 89–114 (2013)
Arosio, L., Bracci, F., Wold, E.F.: Solving the Loewner PDE in complete hyperbolic starlike domains of \(\mathbb{C}^N\). Adv. Math. 242, 209–216 (2013)
Baker, I.N., Pommerenke, C.: On the iteration of analytic functions in a half-plane II. J. London Math. Soc. (2) 20(2), 255–258 (1979)
Bayart, F.: The linear fractional model on the ball. Rev. Mat. Iberoam. 24(3), 765–824 (2008)
Bedford, E.: Open problem session of the Biholomorphic Mappings Meeting at the American Institute of Mathematics. Palo Alto, California (2000)
Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution families and the Loewner Equation I: the unit disc. J. Reine Angew. Math. (Crelle’s J.) 672, 1–37 (2012)
Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution families and the Loewner Equation II: complex hyperbolic manifolds. Math. Ann. 344, 947–962 (2009)
Bracci, F., Gentili, G.: Solving the Schröder equation at the boundary in several variables. Mich. Math. J. 53(2), 337–356 (2005)
Bracci, F., Gentili, G., Poggi-Corradini, P.: Valiron’s construction in higher dimensions. Rev. Mat. Iberoam. 26(1), 57–76 (2010)
Contreras, M.D., Díaz-Madrigal, S., Gumenyuk, P.: Loewner chains in the unit disc. Rev. Mat. Iberoamericana 26, 975–1012 (2010)
Cowen, C.C.: Iteration and the solution of functional equations for functions analytic in the unit disk. Trans. Am. Math. Soc. 265(1), 69–95 (1981)
Docquier, F., Grauert, H.: Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten. Math. Ann. 140, 94–123 (1960)
Duren, P., Graham, I., Hamada, H., Kohr, G.: Solutions for the generalized Loewner differential equation in several complex variables. Math. Ann. 347(2), 411–435 (2010)
Fornæss, J.E.: An increasing sequence of Stein Manifolds whose limit is not Stein. Math. Ann. 223, 275–277 (1976)
Fornæss, J.E.: Short \(\mathbb{C}^k\). Mathematical Society of Japan, Advanced Studies in Pure Mathematics (2003)
Fornæss, J.E., Sibony, N.: Increasing sequences of complex manifolds. Math. Ann. 255(3), 351–360 (1981)
Fornæss, J.E., Stensønes, B.: Stable manifolds of holomorphic hyperbolic maps. Int. J. Math. 15(8), 749–758 (2004)
Graham, I., Hamada, H., Kohr, G.: Parametric representation of univalent mappings in several complex variables. Can. J. Math. 54, 324–351 (2002)
Graham, I., Hamada, H., Kohr, G., Kohr, M.: Asymptotically spirallike mappings in several complex variables. J. Anal. Math. 105, 267–302 (2008)
Jonsson, M., Varolin, D.: Stable manifolds of holomorphic diffeomorphisms. Invent. Math. 149, 409–430 (2002)
Jury, T.: Valiron’s theorem in the unit ball and spectra of composition operators. J. Math. Anal. Appl. 368(2), 482–490 (2010)
Königs, G.: Recherches sur les intégrales de certaines équations fonctionnelles. Ann. Sci. École Norm. Sup. 1(3), 3–41 (1884)
Kufarev, P.P.: On one-parameter families of analytic functions. Mat. Sb. 13, 87–118 (1943). in Russian
Loewner, C.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. Math. Ann. 89, 103–121 (1923)
Peters, H.: Perturbed basins. Math. Ann. 337(1), 1–13 (2007)
Peters, H., Smit, I.M.: Adaptive trains for attracting sequences of holomorphic automorphisms. arXiv:1408.0498 (preprint)
Pommerenke, C.: Über die Subordination analytischer Funktionen. J. Reine Angew. Math. 218, 159–173 (1965)
Pommerenke, C.: On the iteration of analytic functions in a half plane. J. London Mat. Soc. 19(2, 3), 439–447 (1979)
Pfaltzgraff, J.A.: Subordination chains and univalence of holomorphic mappings in \(\mathbb{C}^n\). Math. Ann. 210, 55–68 (1974)
Pfaltzgraff, J.A.: Subordination chains and quasiconformal extension of holomorphic maps in \(\mathbb{C}^n\). Ann. Acad. Sci. Fenn. Ser. A I Math. 1, 13–25 (1975)
Rosay, J.P., Rudin, W.: Holomorphic maps from \(\mathbb{C}^n\) to \(\mathbb{C}^n\). Trans. Am. Math. Soc. 310(1), 47–86 (1988)
Schröder, E.: Über unendliche viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870)
Schröder, E.: Über iterirte Functionen. Math. Ann. 3, 296–322 (1870)
Valiron, G.: Sur l’itération des fonctions holomorphes dans un demi-plan. Bull. Sci. Math. 47, 105–128 (1931)
Wold, E.F.: Fatou-Bieberbach domains. Int. J. Math. 16, 1119–1130 (2005)
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Supported by the ERC grant “HEVO - Holomorphic Evolution Equations” n. 277691.
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Arosio, L. (2015). Abstract Basins of Attraction. In: Bracci, F., Byun, J., Gaussier, H., Hirachi, K., Kim, KT., Shcherbina, N. (eds) Complex Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol 144. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55744-9_4
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