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Domain of Attraction for Maps Tangent to the Identity in \(\mathbb {C} ^2\) with Characteristic Direction of Higher Degree

Abstract

We study holomorphic fixed point germs in two complex variables that are tangent to the identity and have a degenerate characteristic direction. We show that if that characteristic direction is also a characteristic direction for higher degree terms, is non-degenerate for a higher degree term, and satisfies some additional properties, then there is a domain of attraction on which points converge to the origin along that direction.

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Notes

  1. Assumptions (1)–(2) imply that [v] must be: a characteristic direction of degree s, of order one in degree \(k+1\) (so \(t=k\)), and non-degenerate of degree \(r+1\) for some \(k<r<s\).

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Acknowledgments

The author would like to thank Laura DeMarco for useful conversations about this paper. The author would also like to thank Jasmin Raissy, Liz Vivas, Marco Abate, and Mattias Jonsson for comments on an earlier draft. Thanks also to the referee who suggested improvements to this paper.

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Correspondence to Sara Lapan.

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Lapan, S. Domain of Attraction for Maps Tangent to the Identity in \(\mathbb {C} ^2\) with Characteristic Direction of Higher Degree. J Geom Anal 26, 2519–2541 (2016). https://doi.org/10.1007/s12220-015-9637-z

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  • DOI: https://doi.org/10.1007/s12220-015-9637-z

Keywords

  • Holomorphic dynamics
  • Complex dynamics
  • Maps tangent to the identity
  • Domain of attraction
  • Dynamical systems
  • Parabolic fixed point

Mathematics Subject Classification

  • Primary 37F10
  • Secondary 32H50