Fixed points of commuting holomorphic mappings other than the Wolff point

Bracci, Filippo

doi:10.1090/S0002-9947-03-03170-2

Fixed points of commuting holomorphic mappings other than the Wolff point
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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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