The linear fractional model on the ball
Frédéric
Bayart
Source: Rev. Mat. Iberoamericana Volume 24, Number 3
(2008), 765-824.
Abstract
Given a holomorphic self-map $\varphi$ of the ball of $\mathbb{C}^N$, we study whether there exists a map $\sigma$ and a linear fractional transformation $A$ such that $\sigma\circ\varphi=A\circ\sigma$. This is an important result when $N=1$ with a great number of applications. We extend this result to the multi-dimensional setting for a large class of maps. Applications to commuting holomorphic self-maps are given.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.rmi/1228834294
Zentralblatt MATH identifier: 05509263
Mathematical Reviews number (MathSciNet): MR2490162
References
Abate, M.: Iteration theory of holomorphic maps on taut manifolds. Research and Lecture Notes in Mathematics. Complex Analysis and Geometry. Mediterranean Press, Rende, 1989
Mathematical Reviews (MathSciNet): MR1098711
Arnold, V. I.: Geometrical methods in the theory of ordinary differential equations. Grundlehren der Mathematischen Wissenschaften 250. Springer-Verlag, New-York, 1983.
Mathematical Reviews (MathSciNet): MR695786
Baker, I. N. and Pommerenke, Ch.: On the iteration of analytic functions in a half-plane II. J. London Math. Soc. (2) 20 (1979), 255-258.
Mathematical Reviews (MathSciNet): MR551452
Digital Object Identifier: doi:10.1112/jlms/s2-20.2.255
Bayart, F.: A class of linear fractional maps of the ball and their composition operators. Adv. Math. 209 (2007), 649-665.
Mathematical Reviews (MathSciNet): MR2296311
Digital Object Identifier: doi:10.1016/j.aim.2006.05.012
Zentralblatt MATH: 1112.32002
Bisi, C. and Bracci, F.: Linear fractional maps of the unit ball: a geometric study. Adv. in Math. 167 (2002), 265-287.
Mathematical Reviews (MathSciNet): MR1906258
Digital Object Identifier: doi:10.1006/aima.2001.2042
Zentralblatt MATH: 1008.32007
Bourdon, P. and Shapiro, J. H.: Cyclic phenomena for composition operators. Mem. Amer. Math. Soc. 125 (1997), no. 596.
Mathematical Reviews (MathSciNet): MR1396955
Bracci, F.: Common fixed points of commuting holomorphic maps in the unit ball of $\mathbbC^n$. Proc. Amer. Math. Soc. 127 (1999), 1133-1141.
Mathematical Reviews (MathSciNet): MR1610920
Digital Object Identifier: doi:10.1090/S0002-9939-99-04903-5
JSTOR: links.jstor.org
Zentralblatt MATH: 0916.58006
Bracci, F., Contreras, M. and Díaz-Madrigal, S.: Classification of semigroups of linear fractional maps in the unit ball. Adv. Math. 208 (2007), no. 1, 318-350.
Mathematical Reviews (MathSciNet): MR2304320
Digital Object Identifier: doi:10.1016/j.aim.2006.02.010
Zentralblatt MATH: 1127.47038
Bracci, F. and Gentili, G.: Solving the Schröder equation at the boundary in several variables. Michigan Math. J. 53 (2005), 337-356.
Mathematical Reviews (MathSciNet): MR2152705
Digital Object Identifier: doi:10.1307/mmj/1123090773
Project Euclid: euclid.mmj/1123090773
Bracci, F. and Poggi-Corradini, P.: On Valiron's theorem. In Future trends in geometric function theory, 39-55. Rep. Univ. Jyväskylä Dep. Math. Stat. 92, Univ. Jyväskylä, Jyväskylä, 2003.
Mathematical Reviews (MathSciNet): MR2058109
Contreras, M., Díaz-Madrigal, S. and Pommerenke, Ch.: Some remarks on the Abel equation in the unit disk. J. London Math. Soc. (2) 75 (2007), 623-634.
Mathematical Reviews (MathSciNet): MR2352725
Digital Object Identifier: doi:10.1112/jlms/jdm013
Zentralblatt MATH: 1138.30018
Cowen, C.: Iteration and the solution of functional equations for functions analytic in the unit disc. Trans. Amer. Math. Soc. 265 (1981), 69-95.
Mathematical Reviews (MathSciNet): MR607108
Digital Object Identifier: doi:10.2307/1998482
JSTOR: links.jstor.org
Zentralblatt MATH: 0476.30017
Cowen, C. and MacCluer, B.: Linear fractional maps of the ball and their composition operators. Acta. Sci. Math. (Szeged), 66 (2000), 351-376.
Mathematical Reviews (MathSciNet): MR1768872
Cowen, C. and MacCluer, B.: Schroeder's equation in several variables. Taiwanese J. Math. 7 (2003), no. 1, 129-154.
Mathematical Reviews (MathSciNet): MR1961044
Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.: Higher transcendental functions. Vol I. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.
Gallardo-Gutiérrez, E. and Montes-Rodríguez, A.: The role of the spectrum in the cyclic behavior of composition operators. Mem. Amer. Math. Soc. 167 (2004), no. 791.
Mathematical Reviews (MathSciNet): MR2023381
Königs, G.: Recherches sur les intégrales de certaines équations fonctionnelles. Ann. Sci. École Norm. Sup. (3) 1 (1884), 3-41.
MacCluer, B.: Iterates of holomorphic self-maps of the unit ball in $\mathbb C^N$. Michigan Math. J. 30 (1983), 97-106.
Mathematical Reviews (MathSciNet): MR694933
Digital Object Identifier: doi:10.1307/mmj/1029002792
Project Euclid: euclid.mmj/1029002792
Pommerenke, Ch.: On the iteration of analytic functions in a halfplane. J. London Math. Soc. (2) 19 (1979), 439-447.
Mathematical Reviews (MathSciNet): MR540058
Digital Object Identifier: doi:10.1112/jlms/s2-19.3.439
Zentralblatt MATH: 0398.30014
Rudin, W.: Function theory in the unit ball of $\mathbb C^n$. Fundamental Principles of Mathematical Science 241. Springer-Verlag, New York-Berlin, 1980.
Mathematical Reviews (MathSciNet): MR601594
Sternberg, S.: Local contractions and a theorem of Poincaré. Amer. J. Math. 79 (1957), 809-824.
Mathematical Reviews (MathSciNet): MR96853
Digital Object Identifier: doi:10.2307/2372437
JSTOR: links.jstor.org
Ueda, T.: Local structure of analytic transformations of two complex variables. I. J. Math. Kyoto Univ. 26 (1986), 233-261.
Mathematical Reviews (MathSciNet): MR849219
Valiron, G.: Sur l'itération des fonctions holomorphes dans un demi-plan. Bull. Sci. Math. (2) 55 (1931), 105-128.
2012 © Departamento de Matemáticas, Universidad Autónoma de Madrid
Revista Matemática Iberoamericana
