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Perturbed basins of attraction. (English) Zbl 1112.37037
The main result of this paper is the following. Let $$F$$ be an automorphism of $$\mathbb C^k$$ which has an attracting point at $$O$$. Then there exists $$\varepsilon>0$$ such that for any sequence of automorphisms $$\{f_n\}$$ such that $$\max_{\mathbb B^k} \| F-f_n\| <\varepsilon$$ for all $$n$$, the basin of attraction of the random iteration sequence $$\{f_n\circ \ldots \circ f_1\}$$ is biholomorphic to $$\mathbb C^k$$. This type of Fatou-Bieberbach phenomenon is a generalization of J.-P. Rosay and W. Rudin’s [Trans. Am. Math. Soc. 310, 47–86 (1988; Zbl 0708.58003)] and S. Sternberg’s [Am. J. Math. 79, 809–824, (1957; Zbl 0080.29902)] theorem (which works for $$f_n\equiv F$$). The idea of the proof is to find a holomorphic normal form (up to higher-order terms) for which it is easier to prove that the basin of attraction of the random iteration sequence is biholomorphic to $$\mathbb C^k$$. More detailed, the author reduces the previous case to the case of a uniformly attracting sequence of automorphisms (namely such that $$a\| z\| \leq \| f_n(z)\| \leq b\| z\|$$ for all $$z\in \mathbb B^k$$, $$0<a<b<1$$ and $$n\in \mathbb N$$) such that the first jets are triangular with suitably ordered eigenvalues. This last result is related to a theorem by E. F. Wold [Int. J. Math. 16, 1119–1130 (2005; Zbl 1085.32008)].

##### MSC:
 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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##### References:
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