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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Common fixed points of commuting holomorphic maps in the unit ball of ${\mathbb C}^n$


Author: Filippo Bracci
Journal: Proc. Amer. Math. Soc. 127 (1999), 1133-1141
MSC (1991): Primary 32A10, 32A40; Secondary 30E25, 32A30
MathSciNet review: 1610920
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Abstract: Let $\mathbb B^n$ be the unit ball of $\mathbb{C}^n$ ($n>1$). We prove that if $f,g \in \operatorname{Hol}(\mathbb B^n,\mathbb B^n)$ 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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Additional Information

Filippo Bracci
Affiliation: Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Via Belzoni 7, 35131 Padova, Italia
Email: fbracci@math.unipd.it

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04903-5
PII: S 0002-9939(99)04903-5
Keywords: Commuting functions, fixed points, Wolff point
Received by editor(s): July 29, 1997
Communicated by: Steven R. Bell
Article copyright: © Copyright 1999 American Mathematical Society