Abstract
A parabolic cylinder is an invariant, non-recurrent Fatou component \(\Omega \) of an automorphism F of \(\mathbb {C}^2\) satisfying: (1) The closure of the \(\omega \)-limit set of F on \(\Omega \) contains an isolated fixed point, (2) there exists a univalent map \(\Phi \) from \(\Omega \) into \({\mathbb {C}}^2\) conjugating F to the translation \((z,w) \mapsto (z+1, w)\), and (3) every limit map of \(\{F^{\circ n}\}\) on \(\Omega \) has one-dimensional image. In this paper, we prove the existence of parabolic cylinders for an explicit class of maps, and show that examples in this class can be constructed as compositions of shears and overshears.
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Acknowledgements
In a first version of this paper, “parabolic cylinders” were named “non-recurrent Siegel cylinders”. However, the term “parabolic cylinders” seems to be more appropriate, due to Property (2) in Definition 1.1. We thank Eric Bedford for stimulating discussions about this and other facts related to the paper. We also thank the referee for very useful comments which improved much the original paper. In particular, for finding a mistake in the original version of Lemma 2.2.
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Supported by the SIR Grant “NEWHOLITE - New methods in holomorphic iteration” No. RBSI14CFME and by the Research Program P1-0291 from ARRS, Republic of Slovenia
Partially supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006 and PRIN Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics n.2017JZ2SW5.
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Boc Thaler, L., Bracci, F. & Peters, H. Automorphisms of \(\mathbb {C}^2\) with Parabolic Cylinders. J Geom Anal 31, 3498–3522 (2021). https://doi.org/10.1007/s12220-020-00403-4
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DOI: https://doi.org/10.1007/s12220-020-00403-4