Exploding orbits of holomorphic structures

Abstract

Given a vector field on a space, one natural question is how long its flow is defined. In this paper we study time dependant holomorphic vector fields X on C n with holomorphic coefficients. We say that the orbit of p c C n explodes if the integral curve o f p is unbounded on some finite time interval [0, to > C ~÷. Our main question then is whether, in a given class of time dependant holomorphic vector fields, there is a dense subclass of vector fields for each of which a dense set of orbits explode. We study here the cases of time dependant holomorphic Hamiltonian vector fields and holomorphic Reeb vector fields (the time independant case was discussed in ([FG1]) and ([FG2]). In the Hamiltonian case, our main result is Theorem 2.1 in which we obtain density of exploding orbits. To the contrary, in the second case, exploding orbits are not dense. We give two examples: one with a fixed contact form and one with a time dependent contact form. In both these examples, the velocity varies with time.