# American Institute of Mathematical Sciences

2009, 25(2): 537-544. doi: 10.3934/dcds.2009.25.537

## Holomorphic foliations transverse to manifolds with corners

 1 Department of Natural Science - Ryukoku University, Fushimi-Ku, Kyoto 612, Japan 2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21.945-970, Rio de Janeiro, RJ

Received  June 2008 Revised  December 2008 Published  June 2009

We study the geometrical and dynamical properties of a holomorphic vector field on a complex surface, assumed to be transverse to the boundary of a domain which is a non-smooth manifold with boundary and corners. We obtain hyperbolicity and prove a compact leaf result. For a pseudoconvex domain with boundary diffeomorphic to the boundary of a bidisc in $\mathbb C^2$ the foliation is pull-back of a liner hyperbolic foliation. If moreover the diffeomorphism is transversely holomorphic then we have linearization.
Citation: Toshikazu Ito, Bruno Scárdua. Holomorphic foliations transverse to manifolds with corners. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 537-544. doi: 10.3934/dcds.2009.25.537
 [1] Laurence Halpern, Jeffrey Rauch. Hyperbolic boundary value problems with trihedral corners. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4403-4450. doi: 10.3934/dcds.2016.36.4403 [2] Aimin Huang, Roger Temam. The linear hyperbolic initial and boundary value problems in a domain with corners. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1627-1665. doi: 10.3934/dcdsb.2014.19.1627 [3] Doris Bohnet. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. Journal of Modern Dynamics, 2013, 7 (4) : 565-604. doi: 10.3934/jmd.2013.7.565 [4] Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80 [5] M. Jotz. The leaf space of a multiplicative foliation. Journal of Geometric Mechanics, 2012, 4 (3) : 313-332. doi: 10.3934/jgm.2012.4.313 [6] Luis Vega. The dynamics of vortex filaments with corners. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1581-1601. doi: 10.3934/cpaa.2015.14.1581 [7] Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481 [8] Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333 [9] Percy Fernández-Sánchez, Jorge Mozo-Fernández, Hernán Neciosup. Dicritical nilpotent holomorphic foliations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3223-3237. doi: 10.3934/dcds.2018140 [10] Eric Bedford, Serge Cantat, Kyounghee Kim. Pseudo-automorphisms with no invariant foliation. Journal of Modern Dynamics, 2014, 8 (2) : 221-250. doi: 10.3934/jmd.2014.8.221 [11] John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047 [12] Jeffrey J. Early, Juha Pohjanpelto, Roger M. Samelson. Group foliation of equations in geophysical fluid dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1571-1586. doi: 10.3934/dcds.2010.27.1571 [13] Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847 [14] Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1 [15] Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1773-1807. doi: 10.3934/dcds.2013.33.1773 [16] Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443 [17] David DeLatte. Diophantine conditions for the linearization of commuting holomorphic functions. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 317-332. doi: 10.3934/dcds.1997.3.317 [18] Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249 [19] Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557 [20] E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics & Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010

2017 Impact Factor: 1.179