Fixed points of commuting holomorphic mappings other than the Wolff point
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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 $\mathrm {Fix}_A (f):=\{p \in \partial \Delta \mid \lim _{r \to 1}f(rp)=p, \lim _{r \to 1}|f’(rp)|\leq A \}$. 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.References
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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
- MR Author ID: 631111
- Email: fbracci@mat.uniroma2.it
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 1974004