Perturbed basins of attraction.

*(English)*Zbl 1112.37037The 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)].

Reviewer: Filippo Bracci (Roma)

##### 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 |

##### References:

[1] | Bedford, E.: Open problem session of the biholomorphic mappings meeting at the American Institute of Mathematics Palo Alto, CA, 2000 |

[2] | Fornæss, J.E., Stensønes, B.: Stable manifolds of holomorphic hyperbolic maps. Internat. J. Math. 15(8), 749–758 (2004) · Zbl 1071.32016 |

[3] | Jonsson, M., Varolin, D.: Stable manifolds of holomorphic diffeomorphisms. Invent. Math. 149(2), 409–430 (2002) · Zbl 1048.37047 · doi:10.1007/s002220200220 |

[4] | Peters, H., Wold, E.F.: Non-autonomous basins of attraction and their boundaries. J. Geom. Anal. 15, 123–136 (2005) · Zbl 1076.37035 |

[5] | Rosay, J.-P., Rudin, W.: Holomorphic maps from C n to C n , Trans. Amer. Math. Soc. 310(1), 47–86 (1988) · Zbl 0708.58003 |

[6] | Sternberg, S.: Local contractions and a theorem of Poincaré. Amer. J. Math. 79, 809–824 (1957) · Zbl 0080.29902 · doi:10.2307/2372437 |

[7] | Wold, E.F.: Fatou-Bieberbach domains. Internat. J. Math. 16, 1119–1130 (2005) · Zbl 1085.32008 · doi:10.1142/S0129167X05003235 |

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