Common fixed points of commuting holomorphic maps in the unit ball of $\mathbb {C}^{n}$
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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.References
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Additional Information
- Filippo Bracci
- Affiliation: Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Via Belzoni 7, 35131 Padova, Italia
- MR Author ID: 631111
- Email: fbracci@math.unipd.it
- Received by editor(s): July 29, 1997
- Communicated by: Steven R. Bell
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1133-1141
- MSC (1991): Primary 32A10, 32A40; Secondary 30E25, 32A30
- DOI: https://doi.org/10.1090/S0002-9939-99-04903-5
- MathSciNet review: 1610920