Low regularity dynamics are systems of PDEs with initial data and potentials or noises. We assume that both initial data and potentials are of low regularity. Then, one wants to prove well-posedness for this system and provide an approximation of the solution minimising the regularity of the initial data and the given potentials. It turns out that these two procedures rely on iterated integrals expansion of the solution where decorated trees are used for dealing with the complexity of such expansion. We will present two main examples one coming from singular SPDEs for well-posedness and the other which implements a resonance scheme for dispersive equations.
