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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:
[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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