  For a smooth near identity map, we introduce the notion of an interpolating
vector field written in terms of iterates of the map. Our construction is based
on Lagrangian interpolation and provides an explicit expressions for autonomous
vector fields which approximately interpolate the map. We study properties of
the interpolating vector fields and explore their applications to the study of
dynamics. In particular, we construct adiabatic invariants for symplectic near
identity maps. We also introduce the notion of a Poincar\'e section for a near
identity map and use it to visualise dynamics of four dimensional maps. We
illustrate our theory with several examples, including the Chirikov standard
map and a symplectic map in dimension four, an example motivated by the theory
of Arnold diffusion.
