  In this paper we study smooth projective varieties and polarized pairs with
an action of a one dimensional complex torus. As a main tool, we define
birational geometric counterparts of these actions, that, under certain
assumptions, encode the information necessary to reconstruct them. In
particular, we consider some cases of actions of low complexity -- measured in
terms of two invariants of the action, called bandwidth and bordism rank -- and
discuss how they are determined by well known birational transformations,
namely Atiyah flips and Cremona transformations.
