Boundary Behaviour and Linear Fractional Models for Holomorphic Self-Maps of the Unit Ball
by
Graziano Gentili
Universitá di Firenze, Italy

Let f be a holomorphic map from the open unit ball B of Cⁿ into itself, having its Wolff point t at the boundary of B. The talk will present issues on how the behaviour and the geometry of f and of its iterates depend on the differential df_t. In the one dimensional case, the non tangential derivative f^¢(t) of f at the Wolff point always exists and is such that 0 < f^¢(t) £ 1. The case in which f^¢(t) < 1 has been already widely investigated, also in connection with the study of the family of maps which commute with f with respect to composition [6]. The case in which f^¢(t)=1 is still open for many questions, and some new results have been recently obtained, in particular for what concerns the study of the maps which commute with f under composition ([1, 3]). A fundamental theorem due to Cowen ([5, 6]) - in which he constructs a model to represent holomorphic maps of the unit disc D by means of linear fractional transformations - is a main tool in the one variable setting. In the (one and) several variables framework, nice Schwarz-type and Cartan-type theorems at points of the boundary of B have been proved ([4, 9]), and these results have also been generalized and used for the study of identity principles for commuting holomorphic maps ([10, 3]). Moreover, results on how the differential df_t of f at the Wolff point t determines the geometric boundary behaviour of f have been obtained with the use of complex geodesic curves of the unit ball B ([2]). The problem of the construction of linear fractional models in several variables is also a problem of genuine interest, under investigation at the moment, on which new results have been obtained quite recently ([7, 8]).

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[2]

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[3]

F. Bracci, R. Tauraso and F. Vlacci, Identity principles for commuting holomorphic self-maps of the unit disc, Preprint, (1999), Università di Firenze.

[4]

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[5]

C. C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc., 265 (1981), 69-95.

[6]

C. C. Cowen, Commuting analytic functions, Trans. Amer. Math. Soc., 283 (1984), 685-695.

[7]

C. C. Cowen and B. D. MacCluer, Linear fractional maps of the ball and their composition operators, to appear in: Acta Sci. Math. (Szeged).

[8]

C. C. Cowen and B. D. MacCluer, Schroeder's equation in several variables, Preprint, (1999).

[9]

G. Gentili and S. Migliorini, A Boundary Rigidity Problem for Holomorphic Mappings, in: Proceedings of the Third International Workshop on Differential Geometry and its Applications, Sibiu. (1999).

[10]

R. Tauraso and F. Vlacci, Rigidity at the boundary for holomorphic self maps of the unit disc, Preprint, Università di Firenze.

Date received: June 15, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-24.