Abstract
If f is a parabolic holomorphic self-map of the unit disc \(\mathbb {D}\subset \mathbb {C}\), then either for every point z one has \(\lim _{n\to \infty } k_{\mathbb {D}}(f^{n+1}(z),f^n(z))>0\), or the Poincaré distance of any two f-orbits converges to zero. It is an open question whether such a dichotomy holds in the unit ball \(\mathbb {B}^q\subset \mathbb {C}^q\). We show how this question is related to the theory of canonical Kobayashi hyperbolic semi-models, to commuting holomorphic self-maps of the ball and to a purely geometric problem about biholomorphisms of the ball.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Arosio, L.: The stable subset of a univalent self-map. Math. Z. 281(3–4), 1089–1110 (2015)
Arosio, L.: Canonical models for the forward and backward iteration of holomorphic maps. J. Geom. Anal. 27(2), 1178–1210 (2017)
Arosio, L., Bracci, F.: Canonical models for holomorphic iteration. Trans. Am. Math. Soc. 368(5), 3305–3339 (2016)
Arosio, L., Bracci, F.: Simultaneous models for commuting holomorphic self-maps of the ball. Adv. Math. 321, 486–512 (2017)
Arosio, L., Gumenyuk, P.: Valiron and Abel equations for holomorphic self-maps of the polydisc. Int. J. Math. 27(4) (2016)
Baker, I.N., Pommerenke, C.: On the iteration of analytic functions in a half-plane II. J. Lond. 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)
Bracci, F., Gentili, G.: Solving the Schröder equation at the boundary in several variables. Michigan 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)
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)
Cowen, C.C.: Commuting analytic functions. Trans. Am. Math. Soc. 283, 685–695 (1984)
de Fabritiis, C., Gentili, G.: On holomorphic maps which commute with hyperbolic automorphisms. Adv. Math. 144(2), 119–136 (1999)
Fornæss, J.E., Sibony, N.: Increasing sequences of complex manifolds. Math. Ann. 255(3), 351–360 (1981)
Hervé, M.: Quelques propriétés des applications analytiques d’une boule à m dimensions dans elle-même. J. Math. Pures Appl. 42, 117–147 (1963)
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. (3) 1, 3–41 (1884)
Ostapyuk, O.: Backward iteration in the unit ball. Ill. J. Math. 55(4), 1569–1602 (2011)
Pommerenke, C.: On the iteration of analytic functions in a half plane. J. Lond. Mat. Soc. (2) 19(3), 439–447 (1979)
Rosay, J.P., Rudin, W.: Holomorphic maps from \(\mathbb C^n\) to \(\mathbb C^n\). Trans. Am. Math. Soc. 310(1), 47–86 (1988)
Valiron, G.: Sur l’itération des fonctions holomorphes dans un demi-plan. Bull. Sci. Math. 47, 105–128 (1931)
Acknowledgements
This work was supported by the SIR grant “NEWHOLITE—New methods in holomorphic iteration” n. RBSI14CFME.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Arosio, L. (2017). On Parabolic Dichotomy. In: Bracci, F. (eds) Geometric Function Theory in Higher Dimension. Springer INdAM Series, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-73126-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-73126-1_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-73125-4
Online ISBN: 978-3-319-73126-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)