On Fatou-Julia decompositions

— 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. RÉSUMÉ. — Nous proposons une décomposition de Fatou-Julia pour pseudosemigroupes holomorphes. On montre que les ensembres limites des groupes kleiniens de génération finis, les ensembres de Julia d’itérations d’applications et ceux pour des feuilletages réguliers, transversalement holomorphes de codimension complexe un sont des cas particulier de la décomposition. La décomposition est appliquée pour introduire une décomposition de Fatou-Julia pour des feuilletages holomorphes singuliers. Dans les cas bien étudiés, le comportement de la décomposition est comme attendu.

[1]  J. Lehner.  Discontinuous Groups and Automorphic Functions , 1964 .

[2]  C. Loewner.  On semigroups in analysis and geometry , 1964 .

[3]  R. Bott,et al.  Singularities of holomorphic foliations , 1972 .

[4]  R. Schrader,et al.  Mathematical Problems in Theoretical Physics , 1982 .

[5]  G. Sallet,et al.  A sufficient condition for the transitivity of pseudo-semigroups: Application to system theory , 1983 .

[6]  David G. Sullivan,et al.  Solution of the Fatou - Julia problem on wandering domains , 1985 .

[7]  A. Haefliger.  LEAF CLOSURES IN RIEMANNIAN FOLIATIONS , 1988 .

[8]  É. Ghys.  Flots transversalement affines et tissus feuilletés , 1991 .

[9]  N. Sibony,et al.  Complex dynamics in higher dimension , 1993 .

[10]  T. Ueda.  Fatou sets in complex dynamics on projective spaces , 1994 .

[11]  Toshikazu Ito.  A POINCARE-BENDIXSON TYPE THEOREM FOR HOLOMORPHIC VECTOR FIELDS(Singularities of Holomorphic Vector Fields and Related Topics) , 1994 .

[12]  T. Suwa.  Residues of complex analytic foliations relative to singular invariant subvarieties , 1994 .

[13]  Thomas Ransford,et al.  Potential Theory in the Complex Plane: Bibliography , 1995 .

[14]  Aimo H J Hinkkanen,et al.  The Dynamics of Semigroups of Rational Functions I , 1996 .

[15]  S. Yau,et al.  Residues of complex analytic foliations relative to singular invariant subvarieties , 1997 .

[16]  松崎 克彦,et al.  Hyperbolic manifolds and Kleinian groups , 1998 .

[17]  James W. Anderson,et al.  HYPERBOLIC MANIFOLDS AND KLEINIAN GROUPS (Oxford Mathematical Monographs) , 1999 .

[18]  J. Milnor.  Dynamics in one complex variable , 2000 .

[19]  S. Bullett,et al.  Regular and limit sets for holomorphic correspondences , 2001 .

[20]  Hiroki Sumi.  Dimensions of Julia sets of expanding rational semigroups , 2004, math/0405522.

[21]  Yohsuke Takaoka,et al.  On existence of models for the logical system MPCL , 2009 .

[22]  Masahiro Yamamoto,et al.  Hybridized discontinuous Galerkin method with lifting operator , 2010 .

[23]  Tomohiko Ishida.  SECOND COHOMOLOGY CLASSES OF THE GROUP OF C-FLAT DIFFEOMORPHISMS OF THE LINE , 2010 .

[24]  T. Asuke.  A FATOU-JULIA DECOMPOSITION OF TRANSVERSALLY HOLOMORPHIC FOLIATIONS , 2010 .

[25]  Kazufumi Ito,et al.  A Regularization Parameter for Nonsmooth Tikhonov Regularization , 2011, SIAM J. Sci. Comput..

[26]  Norikazu Saito,et al.  ERROR ANALYSIS OF A CONSERVATIVE FINITE-ELEMENT APPROXIMATION FOR THE KELLER-SEGEL SYSTEM OF CHEMOTAXIS , 2010 .

[27]  Yukihiro Seki,et al.  On exact dead-core rates for a semilinear heat equation with strong absorption , 2011 .

[28]  T. Igarashi,et al.  Existence of global solutions in time for reaction-diffusion systems with inhomogeneous terms in cones , 2012 .

[29]  Keiju Sono.  Matrix Coefficients with Minimal K-Types of the Spherical and Non-Spherical Principal Series Representations of SL(3, R) , 2012 .

[30]  S. Kusuoka.  A remark on Malliavin Calculus : Uniform Estimates and Localization , 2013 .