Abstract
Hyperbolicity played an important role in the classification of Fatou components for rational functions in the Riemann sphere. In higher dimensions Fatou components are not nearly as well understood. We investigate the Kobayashi completeness and tautness of invariant Fatou components for holomorphic endomorphisms of ℙ2 and for Henon mappings. We show that basins of attraction and domains with an attracting Riemann surface, previously known to be taut, are also complete, which is strictly stronger. We also prove tautness for a class of Siegel domains.
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