A Fatou-Julia decomposition of transversally holomorphic foliations

Taro Asuke[1]

  • [1] University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba, Meguro-ku Tokyo 153-8914 (Japan)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 3, page 1057-1104
  • ISSN: 0373-0956

Abstract

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A Fatou-Julia decomposition of transversally holomorphic foliations of complex codimension one was given by Ghys, Gomez-Mont and Saludes. In this paper, we propose another decomposition in terms of normal families. Two decompositions have common properties as well as certain differences. It will be shown that the Fatou sets in our sense always contain the Fatou sets in the sense of Ghys, Gomez-Mont and Saludes and the inclusion is strict in some examples. This property is important when discussing a version of Duminy’s theorem in relation to secondary characteristic classes. The structure of Fatou sets is studied in detail, and some properties of Julia sets are discussed. Some similarities and differences between the Julia sets of foliations and those of mapping iterations will be shown. An application to the study of the transversal Kobayashi metrics is also given.

How to cite

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Asuke, Taro. "A Fatou-Julia decomposition of transversally holomorphic foliations." Annales de l’institut Fourier 60.3 (2010): 1057-1104. <http://eudml.org/doc/116289>.

@article{Asuke2010,
abstract = {A Fatou-Julia decomposition of transversally holomorphic foliations of complex codimension one was given by Ghys, Gomez-Mont and Saludes. In this paper, we propose another decomposition in terms of normal families. Two decompositions have common properties as well as certain differences. It will be shown that the Fatou sets in our sense always contain the Fatou sets in the sense of Ghys, Gomez-Mont and Saludes and the inclusion is strict in some examples. This property is important when discussing a version of Duminy’s theorem in relation to secondary characteristic classes. The structure of Fatou sets is studied in detail, and some properties of Julia sets are discussed. Some similarities and differences between the Julia sets of foliations and those of mapping iterations will be shown. An application to the study of the transversal Kobayashi metrics is also given.},
affiliation = {University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba, Meguro-ku Tokyo 153-8914 (Japan)},
author = {Asuke, Taro},
journal = {Annales de l’institut Fourier},
keywords = {Holomorphic foliations; Fatou set; Julia set; Riemannian foliations; holomorphic foliation; Riemannian foliation},
language = {eng},
number = {3},
pages = {1057-1104},
publisher = {Association des Annales de l’institut Fourier},
title = {A Fatou-Julia decomposition of transversally holomorphic foliations},
url = {http://eudml.org/doc/116289},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Asuke, Taro
TI - A Fatou-Julia decomposition of transversally holomorphic foliations
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 3
SP - 1057
EP - 1104
AB - A Fatou-Julia decomposition of transversally holomorphic foliations of complex codimension one was given by Ghys, Gomez-Mont and Saludes. In this paper, we propose another decomposition in terms of normal families. Two decompositions have common properties as well as certain differences. It will be shown that the Fatou sets in our sense always contain the Fatou sets in the sense of Ghys, Gomez-Mont and Saludes and the inclusion is strict in some examples. This property is important when discussing a version of Duminy’s theorem in relation to secondary characteristic classes. The structure of Fatou sets is studied in detail, and some properties of Julia sets are discussed. Some similarities and differences between the Julia sets of foliations and those of mapping iterations will be shown. An application to the study of the transversal Kobayashi metrics is also given.
LA - eng
KW - Holomorphic foliations; Fatou set; Julia set; Riemannian foliations; holomorphic foliation; Riemannian foliation
UR - http://eudml.org/doc/116289
ER -

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