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Zbl 0941.32018
Bracci, Filippo
Commuting holomorphic maps in strongly convex domains. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 27, No.1, 131-144 (1998). ISSN 0391-173X

Summary: Let $D$ be a bounded strongly convex $C^3$ domain of $\bbfC^n$. We prove that if $f,g\in \text {Hol}(D,D)$ are commuting holomorphic self-maps of $D$, then they have a common fixed point in $\overline D$ (if it belongs to $\partial D$, we mean fixed in the sense of $K$-limits). Furthermore, if $f$ and $g$ have no fixed points in $D$ and $f\circ g=g\circ f$ then $f$ and $g$ have the same Wolff point, unless their restrictions to the complex geodesic whose closure contains the Wolff points of $f$ and $g$, are two commuting (hyperbolic) automorphisms of such geodesic.

MSC 2000:: *32H02 Holomorphic mappings on analytic spaces
32H50 Iteration problems for holomorphic maps on analytic spaces
32F45 Invariant metrics and pseudodistances

Keywords: commuting holomorphic maps; strongly convex domains

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