Exploding orbits of holomorphic structures (Q5961420)

Given a vector field on a space, one natural question is how long its flow is defined. In this paper the authors study time dependent holomorphic vector fields \(X\) on \(\mathbb{C}^n\) with holomorphic coefficients. The orbit of \(p\in\mathbb{C}^n\) explodes if the integral curve of \(p\) is unbounded on some finite time interval \([0,t_0]\subset\mathbb{R}^+\). The main question then is whether, in a given class of time dependent holomorphic vector fields, there is a dense subclass of vector fields for each of which a dense set of orbits explode. The authors study here the cases of time dependent holomorphic Hamiltonian vector fields and holomorphic Reeb vector fields (the time independent case was discussed by the authors [C. R. Acad. Sci., Paris, Ser. I 319, No. 6, 553-557 (1994; Zbl 0853.58049)]. In the Hamiltonian case, they obtain density of exploding orbits. To the contrary in the symplectic case, exploding orbits are not dense. They 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.