Common fixed points of commuting holomorphic maps in the unit ball of ℂⁿ

Bracci, Filippo

doi:10.1090/S0002-9939-99-04903-5

Common fixed points of commuting holomorphic maps in the unit ball of $\mathbb {C}^{n}$
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Proc. Amer. Math. Soc. 127 (1999), 1133-1141 Request permission

Abstract:

Let $\mathbb {B}^n$ be the unit ball of $\mathbb {C}^n$ ($n>1$). We prove that if $f,g \in \mathrm {Hol}(\mathbb {B}^n, \mathbb {B}^n)$ are holomorphic self-maps of $\mathbb {B}^n$ such that $f \circ g = g \circ f$, then $f$ and $g$ have a common fixed point (possibly at the boundary, in the sense of $K$-limits). Furthermore, if $f$ and $g$ have no fixed points in $\mathbb {B}^n$, then they have the same Wolff point, unless the restrictions of $f$ and $g$ to the one-dimensional complex affine subset of $\mathbb {B}^n$ determined by the Wolff points of $f$ and $g$ are commuting hyperbolic automorphisms of that subset.

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