Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-25T00:28:36.683Z Has data issue: false hasContentIssue false
  • English
  • Français

A local study near the Wolff point on the ball

Part of: Pseudoconvex domains Holomorphic mappings and correspondences

Published online by Cambridge University Press:  17 June 2020

Fengbai Li  and
Feng Rong
Fengbai Li
Affiliation:
School of Mathematics, Shanghai University of Finance and Economics, 777 Guo Ding Road, Shanghai200433, P.R. China (li.fengbai@mail.shufe.edu.cn)
Feng Rong
Affiliation:
School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai200240, P.R. China (frong@sjtu.edu.cn)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let f be a holomorphic self-map of the unit ball in dimension 2, which does not have an interior fixed point. Suppose that f has a Wolff point p with the boundary dilatation coefficient equal to 1 and the non-tangential differential dfp = id. The local behaviours of f near p are quite diverse, and we give a detailed study in many typical cases. As a byproduct, we give a dynamical interpretation of the Burns–Krantz rigidity theorem. Note also that similar results hold on two-dimensional contractible smoothly bounded strongly pseudoconvex domains.

Keywords

Wolff pointSiegel upper half-planeholomorphic self-map

MSC classification

Primary: 32H50: Iteration problems
Secondary: 32T15: Strongly pseudoconvex domains
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 3 , August 2020 , pp. 761 - 779
Check for updates
Copyright
Copyright © The Authors, 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Abate, M., Horospheres and iterates of holomorphic maps, Math. Z. 198 (1988), 225–238.CrossRefGoogle Scholar
2Abate, M., Iteration theory of holomorphic maps on taut manifolds (Mediterranean Press, Rende, Cosenza, 1989).Google Scholar
3Abate, M., Angular derivatives in strongly pseudoconvex domains, in Proceedings of Symposia in Pure Mathematics, Volume 52 (2), pp. 23–40 (American Mathematical Society, Providence, RI, 1991).CrossRefGoogle Scholar
4Arosio, L., Canonical models for the forward and backward iteration of holomorphic maps, J. Geom. Anal. 27 (2017), 1178–1210.CrossRefGoogle Scholar
5Arosio, L. and Bracci, F., Canonical models for holomorphic iteration, Trans. Amer. Math. Soc. 368 (2016), 3305–3339.CrossRefGoogle Scholar
6Bayart, F., The linear fractional model on the ball, Rev. Mat. Iberoam. 24 (2008), 765–824.CrossRefGoogle Scholar
7Bourdon, P. S. and Shapiro, J. H., Cyclic phenomena for composition operators, Memoirs of the American Mathematical Society, Volume 125, no. 596 (American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
8Bracci, F., Dilatation and order of contact for holomorphic self-maps of strongly convex domains, Proc. Lond. Math. Soc. 86 (2003), 131–152.CrossRefGoogle Scholar
9Bracci, F. and Gentili, G., Solving the Schröder equation at the boundary in several variables, Michigan Math. J. 53 (2005), 337–356.CrossRefGoogle Scholar
10Bracci, F. and Zaitsev, D., On biholomorphisms between bounded quasi-Reinhardt domains, J. Funct. Anal. 254 (2008), 1449–1466.CrossRefGoogle Scholar
11Burns, D. and Krantz, S. G., Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc. 7 (1994), 661–676.CrossRefGoogle Scholar
12Diederich, K., Fornæss, J. E. and Wold, E. F., Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-type, J. Geom. Anal. 24 (2014), 2124–2134.CrossRefGoogle Scholar
13Huang, X., A preservation principle of extremal mappings near a strongly pseudoconvex point and its applications, Illinois J. Math. 38 (1994), 283–302.CrossRefGoogle Scholar
14Ma, D., On iterates of holomorphic maps, Math. Z. 207 (1991), 417–428.CrossRefGoogle Scholar
15Rong, F., A brief survey on local holomorphic dynamics in higher dimensions, in Complex analysis and geometry, Springer Proceedings in Mathematics and Statistics, Volume 144, pp. 295–307 (Springer, Tokyo, 2015).Google Scholar
16Rudin, W., Function theory in the unit ball of ℂn, Grundlehren der mathematischen Wissenschaften, Volume 241 (Springer-Verlag, New York, 1980).CrossRefGoogle Scholar
17Vitushkin, A. G., Holomorphic mappings and the geometry of hypersurfaces, in Introduction to complex analysis, pp. 159–214 (Springer-Verlag, Berlin Heidelberg, 1997).Google Scholar