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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Fixed points of commuting holomorphic mappings other than the Wolff point
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by Filippo Bracci PDF
Trans. Amer. Math. Soc. 355 (2003), 2569-2584 Request permission

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.
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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