Fixed points of commuting holomorphic mappings other than the Wolff point

Author:
Filippo Bracci

Journal:
Trans. Amer. Math. Soc. **355** (2003), 2569-2584

MSC (2000):
Primary 30D05; Secondary 30C80, 30E25, 47B33

DOI:
https://doi.org/10.1090/S0002-9947-03-03170-2

Published electronically:
January 29, 2003

MathSciNet review:
1974004

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the unit disc of and let be such that . For , let . We study the behavior of on . In particular, we prove that . As a consequence, besides conditions for , we prove a conjecture of C. Cowen in case and are univalent mappings.

**[Ab]**M. Abate,*Iteration theory of holomorphic maps on taut manifolds*. Mediterranean Press, Rende, Italy 1989. MR**92i:32032****[Be]**D.F. Behan,*Commuting analytic functions without fixed points.*Proc. Amer. Math. Soc. 37 (1973), 114-120. MR**46:7492****[Br1]**F. Bracci,*Common fixed points of commuting holomorphic maps in the unit ball of*. Proc. Amer. Math. Soc. 127, 4, (1999), 1133-1141. MR**99f:32034****[Br2]**F. Bracci,*Fixed points of commuting holomorphic maps without boundary regularity*. Canad. Math. Bull. 43, 3, (2000), 294-303. MR**2001g:32039****[ChMo]**I. Chalendar and R. Mortini,*When do finite Blaschke products commute?*. Bull. Australian Math. Soc., 64 (2001), 189-200. MR**2002i:30039****[Co1]**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. MR**82i:30036****[Co2]**C.C. Cowen,*Commuting analytic functions*. Trans. Amer. Math. Soc. 283,2, (1984), 685-695. MR**85i:30054****[CoPo]**C.C. Cowen - Ch. Pommerenke,*Inequalities for the angular derivative of an analytic function in the unit disk*. J. London Math. Soc. (2), 26 (1982), 271- 289. MR**84a:30006****[Go]**G. M. Goluzin,*Geometric theory of functions of a complex variable*. Transl. of Math. Monographs, 26, Amer. Math. Soc. 1969. MR**40:308****[PC1]**P. Poggi Corradini,*Angular derivatives at boundary fixed points for self-maps of the disk*. Proc. Amer. Math. Soc. 126, 6, (1998), 1697-1708. MR**98g:30049****[PC2]**P. Poggi Corradini,*Canonical conjugations at fixed points other than the Denjoy-Wolff point*. Ann. Acad. Sci. Fenn. Math. 25 (2000), 2, 487-499. MR**2001f:30033****[PC3]**P. Poggi Corradini,*Backward sequences with bounded hyperbolic steps for analytic self-maps of the disk*. Revista Matematica Iberoamericana, to appear.**[Po]**C. Pommerenke,*Boundary behaviour of conformal maps*. Springer-Verlag, New York, 1992. MR**95b:30008****[Sha]**J.H. Shapiro,*Composition operators and classical function theory*. Springer-Verlag, New York, 1993. MR**94k:47049****[Shi]**A. L. Shields,*On fixed points of commuting analytic functions*. Proc. Amer. Math. Soc. 15 (1964), 703-706. MR**29:2790**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
30D05,
30C80,
30E25,
47B33

Retrieve articles in all journals with MSC (2000): 30D05, 30C80, 30E25, 47B33

Additional Information

**Filippo Bracci**

Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy

Email:
fbracci@mat.uniroma2.it

DOI:
https://doi.org/10.1090/S0002-9947-03-03170-2

Keywords:
Fixed points; Wolff point; commuting mappings

Received by editor(s):
April 1, 2001

Published electronically:
January 29, 2003

Additional Notes:
Partially supported by Progetto MURST di Rilevante Interesse Nazionale Proprietà geometriche delle varietà reali e complesse and GNSAGA

Article copyright:
© Copyright 2003
American Mathematical Society