Flow maps: take for example an autonomous ODE x'(t) = f(x(t)) f(.) describes a vector field, right? A flow map is a function w_t(u) = x(t) that solves the

Flow maps: take for example an autonomous ODE x'(t) = f(x(t)) f(.) describes a vector field, right? A flow map is a function w_t(u) = x(t) that solves the ODE with x(0) = u, for fixed t. If f(.) is invertible, then each flow map is a group in the very same way rotations of a Rubik cube are a group. If not, it's a semigroup, which is a group without an invertibility. The former describes systems that you can track back in time and calculate initial conditions only from looking at the present state. My dissertation was actually about numerically integrating symplectic vector fields; I spent a lot of time hunching over Arnold's "Mathematical methods of classical mechanics".