Iterates of holomorphic self-maps on pseudoconvex domains of finite and infinite type in $\mathbb C^n$
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- by Tran Vu Khanh and Ninh Van Thu PDF
- Proc. Amer. Math. Soc. 144 (2016), 5197-5206 Request permission
Abstract:
Using the lower bounds on the Kobayashi metric established by the first author, we prove a Wolff-Denjoy-type theorem for a very large class of pseudoconvex domains in $\mathbb {C}^n$. This class includes many pseudoconvex domains of finite type and infinite type.References
- Marco Abate, Boundary behaviour of invariant distances and complex geodesics, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 80 (1986), no. 3, 100–106 (1987). MR 976695
- 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, compactly divergent sequences and commuting holomorphic maps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), no. 2, 167–191. MR 1129300
- 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 Jasmin Raissy, Wolff-Denjoy theorems in nonsmooth convex domains, Ann. Mat. Pura Appl. (4) 193 (2014), no. 5, 1503–1518. MR 3262645, DOI 10.1007/s10231-013-0341-y
- David Catlin, Necessary conditions for subellipticity of the $\bar \partial$-Neumann problem, Ann. of Math. (2) 117 (1983), no. 1, 147–171. MR 683805, DOI 10.2307/2006974
- David Catlin, Subelliptic estimates for the $\overline \partial$-Neumann problem on pseudoconvex domains, Ann. of Math. (2) 126 (1987), no. 1, 131–191. MR 898054, DOI 10.2307/1971347
- Sanghyun Cho, A lower bound on the Kobayashi metric near a point of finite type in $\textbf {C}^n$, J. Geom. Anal. 2 (1992), no. 4, 317–325. MR 1170478, DOI 10.1007/BF02934584
- A. Denjoy, Sur l’itération des fonctions analytiques, C. R. Acad. Sci. Paris 182(1926), 255–257.
- Klas Diederich and John E. Fornæss, Proper holomorphic maps onto pseudoconvex domains with real-analytic boundary, Ann. of Math. (2) 110 (1979), no. 3, 575–592. MR 554386, DOI 10.2307/1971240
- Franc Forstnerič and Jean-Pierre Rosay, Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann. 279 (1987), no. 2, 239–252. MR 919504, DOI 10.1007/BF01461721
- Hervé Gaussier, Tautness and complete hyperbolicity of domains in $\textbf {C}^n$, Proc. Amer. Math. Soc. 127 (1999), no. 1, 105–116. MR 1458872, DOI 10.1090/S0002-9939-99-04492-5
- Michel Hervé, Quelques propriétés des applications analytiques d’une boule à $m$ dimensions dan elle-même, J. Math. Pures Appl. (9) 42 (1963), 117–147 (French). MR 159962
- Xiao Jun Huang, A non-degeneracy property of extremal mappings and iterates of holomorphic self-mappings, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 3, 399–419. MR 1310634
- Anders Karlsson, On the dynamics of isometries, Geom. Topol. 9 (2005), 2359–2394. MR 2209375, DOI 10.2140/gt.2005.9.2359
- Tran Vu Khanh, Lower bounds on the Kobayashi metric near a point of infinite type, J. Geom. Anal. 26 (2016), no. 1, 616–629. MR 3441530, DOI 10.1007/s12220-015-9565-y
- Tran Vu Khanh and Giuseppe Zampieri, Regularity of the $\overline \partial$-Neumann problem at point of infinite type, J. Funct. Anal. 259 (2010), no. 11, 2760–2775. MR 2719273, DOI 10.1016/j.jfa.2010.08.004
- Jeffery D. McNeal, Convex domains of finite type, J. Funct. Anal. 108 (1992), no. 2, 361–373. MR 1176680, DOI 10.1016/0022-1236(92)90029-I
- Peter R. Mercer, Complex geodesics and iterates of holomorphic maps on convex domains in $\textbf {C}^n$, Trans. Amer. Math. Soc. 338 (1993), no. 1, 201–211. MR 1123457, DOI 10.1090/S0002-9947-1993-1123457-0
- R. Michael Range, The Carathéodory metric and holomorphic maps on a class of weakly pseudoconvex domains, Pacific J. Math. 78 (1978), no. 1, 173–189. MR 513293
- Tomasz Warszawski, Boundary behavior of the Kobayashi distance in pseudoconvex Reinhardt domains, Michigan Math. J. 61 (2012), no. 3, 575–592. MR 2975263, DOI 10.1307/mmj/1347040260
- J. Wolff, Sur l’iteration des fonctions bornes, C. R. Acad. Sci. Paris 182(1926), 200–201.
- Wenjun Zhang and Fyuao Ren, Dynamics on weakly pseudoconvex domains, Chinese Ann. Math. Ser. B 16 (1995), no. 4, 467–476. A Chinese summary appears in Chinese Ann. Math. Ser. A 16 (1995), no. 6, 798. MR 1380584
Additional Information
- Tran Vu Khanh
- Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, NSW, Australia, 2522
- MR Author ID: 815734
- Email: tkhanh@uow.edu.au
- Ninh Van Thu
- Affiliation: Department of Mathematics, Vietnam National University at Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
- MR Author ID: 853151
- Email: thunv@vnu.edu.vn
- Received by editor(s): July 16, 2015
- Received by editor(s) in revised form: December 25, 2015, December 28, 2015, January 13, 2016, and February 4, 2016
- Published electronically: May 23, 2016
- Additional Notes: The research of the first author was supported by the Australian Research Council DE160100173.
The research of the second author was supported by the Vietnam National University, Hanoi (VNU) under project number QG.16.07. This work was completed when the second author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for the financial support and hospitality. - Communicated by: Franc Forstneric
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 5197-5206
- MSC (2010): Primary 32H50; Secondary 37F99
- DOI: https://doi.org/10.1090/proc/13138
- MathSciNet review: 3556264