We present a simple rational map of the complex projective plane whose first and second dynamical degrees coincide, but which does not have any invariant foliation.

We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques.

where HJ = HJ(x, y, t) is a polynomial of x and y of which the coefficients are rational functions of t holomorphic in BJ := P−ΞJ , ΞJ 3 ∞ being the fixed singular points of the J-th Painlevé equation (Ref. 2). Here the equivalece means that the second order nonlinear differential equation in x obtained from the J-th Painlevé system by elimination of y is just the J-th Painlevé equation. The J-...

Abstract We give conditions for a conformal vector field to be tangent null hypersurface. particularize two important cases: Killing and closed field. In the first case, we obtain result ensuring that hypersurface is horizon. second one, gives rise foliation of manifold by totally umbilical hypersurfaces with constant mean curvature which can spacelike, timelike or null. prove several results e...