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**Common fixed points of commuting holomorphic maps in the unit ball of \({\mathbb C}^n\).**
*(English)*
Zbl 0916.58006

In the papers [D. F. Behan, Proc. Am. Math. Soc. 37, 114-120 (1973; Zbl 0251.30009) and A. L. Shields, Proc. Am. Math. Soc. 15, 703-706 (1964; Zbl 0129.29104)], Behan and Shields showed that, except for the case of two hyperbolic automorphisms of \(\Delta\), the unit disk of \(\mathbb{C}\), two non-trivial commuting holomorphic self-maps of \(\Delta\) have the same fixed point in \(\Delta\) or the same Wolff point in \(\partial\Delta\).

In this paper, under the only hypothesis that \(f,g \in \text{Hol}(\mathbb{B} ^n,\mathbb{B} ^n)\) are such that \(f\circ g = g\circ f\), the author proves that either there exists \(z_0\in\mathbb{B} ^n\) such that \(f(z_0) = g(z_0) =z_0\), or there exists \(\tau\in\partial\mathbb{B} ^n\) such that K-\(\lim_{z\to\tau} f(z) =\) K-\(\lim_{z\to\tau} g(z) =\tau\). If the maps have no fixed points, an upper bound for the boundary dilatation coefficients at the Wolff point is established. A simple example of commuting maps with different Wolff points which are not automorphisms of \(\mathbb{B} ^n, n > 1\), but whose first components are indeed hyperbolic automorphisms of \(\Delta\) is constructed. This is in a certain sense the only case in which two commuting holomorphic self-maps of \(\mathbb{B} ^n\) may have different Wolff points. A Behan Shields-type theorem holds in \(\mathbb{B} ^n\): two commuting holomorphic maps with no fixed points either have the same Wolff point or they are conjugated to two commuting holomorphic maps whose first components are hyperbolic automorphisms of \(\Delta\).

In this paper, under the only hypothesis that \(f,g \in \text{Hol}(\mathbb{B} ^n,\mathbb{B} ^n)\) are such that \(f\circ g = g\circ f\), the author proves that either there exists \(z_0\in\mathbb{B} ^n\) such that \(f(z_0) = g(z_0) =z_0\), or there exists \(\tau\in\partial\mathbb{B} ^n\) such that K-\(\lim_{z\to\tau} f(z) =\) K-\(\lim_{z\to\tau} g(z) =\tau\). If the maps have no fixed points, an upper bound for the boundary dilatation coefficients at the Wolff point is established. A simple example of commuting maps with different Wolff points which are not automorphisms of \(\mathbb{B} ^n, n > 1\), but whose first components are indeed hyperbolic automorphisms of \(\Delta\) is constructed. This is in a certain sense the only case in which two commuting holomorphic self-maps of \(\mathbb{B} ^n\) may have different Wolff points. A Behan Shields-type theorem holds in \(\mathbb{B} ^n\): two commuting holomorphic maps with no fixed points either have the same Wolff point or they are conjugated to two commuting holomorphic maps whose first components are hyperbolic automorphisms of \(\Delta\).

Reviewer: Eleonora Storozhenko (Odessa)

### MSC:

58C30 | Fixed-point theorems on manifolds |