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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
Published electronically: January 29, 2003
MathSciNet review: 1974004
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Abstract: Let $\Delta$ be the unit disc of $\mathbb C$ and let $f,g \in \mathrm{Hol}(\Delta,\Delta)$ be such that $f \circ g = g \circ f$. For $A>1$, let $\mathrm{Fix}_A (f):=\{p \in \partial\Delta \mid \lim_{r \to 1}f(rp)=p, \lim_{r \to 1}\vert f'(rp)\vert\leq A \}$. We study the behavior of $g$ on $\mathrm{Fix}_A (f)$. In particular, we prove that $g(\mathrm{Fix}_A (f))\subseteq \mathrm{Fix}_A (f)$. As a consequence, besides conditions for $\mathrm{Fix}_A(f) \cap \mathrm{Fix}_A(g) \neq \emptyset$, we prove a conjecture of C. Cowen in case $f$ and $g$ are univalent mappings.

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Additional Information

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

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

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