Abstract
Fatou components for rational endomorphisms of the Riemann sphere are fully classified and play an important role in our view of one-dimensional dynamics. In higher dimensions, the situation is less satisfactory. In this work we give a nearly complete classification of invariant Fatou components for moderately dissipative Hénon maps. Namely, we prove that any such a component is either an attracting or parabolic basin, or the basin of a rotation domain. More specifically, recurrent Fatou components were classified about 20 years ago (modulo the problem of existence of Herman ring basins), while in this paper we prove that non-recurrent invariant Fatou components are semi-parabolic basins. Most of our methods apply in a more general setting.
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Lyubich, M., Peters, H. Classification of invariant Fatou components for dissipative Hénon maps. Geom. Funct. Anal. 24, 887–915 (2014). https://doi.org/10.1007/s00039-014-0280-9
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DOI: https://doi.org/10.1007/s00039-014-0280-9