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On Fatou-Julia decompositions
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 1, pp. 155-195.

We propose a Fatou-Julia decomposition for holomorphic pseudosemigroups. It will be shown that the limit sets of finitely generated Kleinian groups, the Julia sets of mapping iterations and Julia sets of complex codimension-one regular foliations can be seen as particular cases of the decomposition. The decomposition is applied in order to introduce a Fatou-Julia decomposition for singular holomorphic foliations. In the well-studied cases, the decomposition behaves as expected.

Nous proposons une décomposition de Fatou-Julia pour les pseudosemigroupes holomorphes. On montre que les ensembles limites des groupes kleiniens de type fini, les ensembles de Julia de fonctions holomorphes et ceux des feuilletages réguliers transversalement holomorphes de codimension un sont des cas particuliers de cette décomposition. La décomposition est utilisée pour introduire une décomposition de Fatou-Julia pour les feuilletages holomorphes singuliers. Dans les cas étudiés, le comportement de la décomposition est comme attendu.

Published online:
DOI: 10.5802/afst.1369
Taro Asuke 1

1 Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
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Taro Asuke. On Fatou-Julia decompositions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 1, pp. 155-195. doi : 10.5802/afst.1369. https://afst.centre-mersenne.org/articles/10.5802/afst.1369/

[1] Asuke (T.).— A Fatou-Julia decomposition of transversally holomorphic foliations, Ann. Inst. Fourier (Grenoble) 60, p. 1057-1104 (2010). | Numdam | MR | Zbl

[2] Baum (P.) and Bott (R.).— Singularities of holomorphic foliations, J. Differential Geom. 7, p. 279-342 (1972). | MR | Zbl

[3] Bullett (S.) and Penrose (C.).— Regular and limit sets for holomorphic correspondences, Fund. Math. 167, p. 111-171 (2001). | MR | Zbl

[4] Fornæss (J.) and Sibony (N.).— Complex dynamics in higher dimension I., Complex analytic methods in dynamical systems (Rio de Janeiro, 1992), Astérisque, vol. 222, p. 5, p. 201-231 (1994). | MR | Zbl

[5] Ghys (É.).— Flots transversalement affines et tissus feuilletés, Analyse globale et physique mathématique (Lyon, 1989), Mém. Soc. Math. France (N.S.) 46, p. 123-150 (1991). | Numdam | MR | Zbl

[6] Ghys (É.), Gómez-Mont (X.), and Saludes (J.).— Fatou and Julia Components of Transversely Holomorphic Foliations, Essays on Geometry and Related Topics: Memoires dediés à André Haefliger (É. Ghys, P. de la Harpe, V.F.R. Jones, V. Sergiescu, and T. Tsuboi, eds.), Monogr. Enseign. Math., vol. 38, Enseignement Math., Geneva, p. 287-319 (2001). | MR | Zbl

[7] Haefliger (A.).— Leaf closures in Riemannian foliations, A fête of topology, Academic Press, Boston, MA, p. 3-32 (1988). | MR | Zbl

[8] Haefliger (A.).— Foliations and compactly generated pseudogroups, Foliations: geometry and dynamics (Warsaw, 2000), World Sci. Publ., River Edge, NJ, p. 275-295 (2002). | MR | Zbl

[9] Hinkkanen (A.) and Martin (G.J.).— The dynamics of semigroups of rational functions I, Proc. London Math. Soc. (3) 73, p. 358-384 (1996). | MR | Zbl

[10] Ito (T.).— A Poincaré-Bendixson type theorem for holomorphic vector fields, Singularities of holomorphic vector fields and related topics (Kyoto, 1993), Sūrikaisekikenkyūsho Kōkyūroku, Kyoto Univ. Research Institute for Mathematical Sciences, Kyoto, Japan, p. 1-9 (1994). | MR | Zbl

[11] Kupka (I.) and Sallet (G.).— A sufficient condition for the transitivity of pseudosemigroups: application to system theory, J. Differential Equations 47, p. 462-470 (1983). | MR | Zbl

[12] Lehner (J.).— Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, Amer. Math. Soc., Providence, RI (1964). | MR | Zbl

[13] Loewner (C.).— On semigroups in analysis and geometry, Bull. Amer. Math. Soc. 70, p. 1-15 (1964). | MR | Zbl

[14] Matsuzaki (K.) and Taniguchi (M.).— Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1998). | MR | Zbl

[15] Milnor (J.).— Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ (2006). | MR | Zbl

[16] Morosawa (S.), Nishimura (Y.), Taniguchi (M.), and Ueda (T.).— Holomorphic dynamics, Cambridge Studies in Advanced Mathematics, vol. 66, Cambridge University Press, Cambridge (2000). | MR | Zbl

[17] Ransford (T.).— Potential theory in the complex plane, London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge (1995). | MR | Zbl

[18] Sullivan (D.).— Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122, p. 401-418 (1985). | MR | Zbl

[19] Sumi (H.).— Dimensions of Julia sets of expanding rational semigroups, Kodai Math. J. 28, p. 390-422 (2005). | MR | Zbl

[20] Suwa (T.).— Residues of complex analytic foliations relative to singular invariant subvarieties, Singularities and complex geometry (Beijing, 1994), AMS/IP Stud. Adv. Math., 5, Amer. Math. Soc., Providence, RI, p. 230-245 (1997). | MR | Zbl

[21] Ueda (T.).— Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan 46, p. 545-555 (1994). | MR | Zbl

[22] Woronowicz (S.L.).— Pseudospaces, pseudogroups and Pontriagin duality, Mathematical problems in theoretical physics (Proc. Internat. Conf. Math. Phys., Lausanne, 1979), Lecture Notes in Phys., vol. 116, Springer, Berlin-New York, p. 407-412 (1980). | MR | Zbl

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