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

Interpolating vector fields for near identity maps and averaging