On Fatou-Julia decompositions

Taro Asuke[1]

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

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 1, page 155-195
  • ISSN: 0240-2963

Abstract

top
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.

How to cite

top

Asuke, Taro. "On Fatou-Julia decompositions." Annales de la faculté des sciences de Toulouse Mathématiques 22.1 (2013): 155-195. <http://eudml.org/doc/275354>.

@article{Asuke2013,
abstract = {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.},
affiliation = {Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan},
author = {Asuke, Taro},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {6},
number = {1},
pages = {155-195},
publisher = {Université Paul Sabatier, Toulouse},
title = {On Fatou-Julia decompositions},
url = {http://eudml.org/doc/275354},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Asuke, Taro
TI - On Fatou-Julia decompositions
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 1
SP - 155
EP - 195
AB - 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.
LA - eng
UR - http://eudml.org/doc/275354
ER -

References

top
  1. Asuke (T.).— A Fatou-Julia decomposition of transversally holomorphic foliations, Ann. Inst. Fourier (Grenoble) 60, p. 1057-1104 (2010). Zbl1198.57020MR2680824
  2. Baum (P.) and Bott (R.).— Singularities of holomorphic foliations, J. Differential Geom. 7, p. 279-342 (1972). Zbl0268.57011MR377923
  3. Bullett (S.) and Penrose (C.).— Regular and limit sets for holomorphic correspondences, Fund. Math. 167, p. 111-171 (2001). Zbl0984.37045MR1816043
  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). Zbl0813.58030MR1285389
  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). Zbl0761.57016MR1125840
  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). Zbl1013.37043MR1929331
  7. Haefliger (A.).— Leaf closures in Riemannian foliations, A fête of topology, Academic Press, Boston, MA, p. 3-32 (1988). Zbl0667.57012MR928394
  8. Haefliger (A.).— Foliations and compactly generated pseudogroups, Foliations: geometry and dynamics (Warsaw, 2000), World Sci. Publ., River Edge, NJ, p. 275-295 (2002). Zbl1002.57059MR1882774
  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). Zbl0859.30026MR1397693
  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). Zbl0900.32014MR1332098
  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). Zbl0527.93029MR692840
  12. Lehner (J.).— Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, Amer. Math. Soc., Providence, RI (1964). Zbl0178.42902MR164033
  13. Loewner (C.).— On semigroups in analysis and geometry, Bull. Amer. Math. Soc. 70, p. 1-15 (1964). Zbl0196.23701MR160192
  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). Zbl0892.30035MR1638795
  15. Milnor (J.).— Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ (2006). Zbl1085.30002MR2193309
  16. Morosawa (S.), Nishimura (Y.), Taniguchi (M.), and Ueda (T.).— Holomorphic dynamics, Cambridge Studies in Advanced Mathematics, vol. 66, Cambridge University Press, Cambridge (2000). Zbl0979.37001MR1747010
  17. Ransford (T.).— Potential theory in the complex plane, London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge (1995). Zbl0828.31001MR1334766
  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). Zbl0589.30022MR819553
  19. Sumi (H.).— Dimensions of Julia sets of expanding rational semigroups, Kodai Math. J. 28, p. 390-422 (2005). Zbl1092.37027MR2153926
  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). Zbl0916.32023MR1468279
  21. Ueda (T.).— Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan 46, p. 545-555 (1994). Zbl0829.58025MR1276837
  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). Zbl0513.46046MR582650

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.