  We show that the action of the mapping class group on the space of closed
curves of a closed surface effectively tracks the corresponding action on
Teichm\"uller space in the following sense: for all but quantitatively few
mapping classes, the information of how a mapping class moves a given point of
Teichm\"uller space determines, up to a power saving error term, how it changes
the geometric intersection numbers of a given closed curve with respect to
arbitrary geodesic currents. Applications include an effective estimate
describing the speed of convergence of Teichm\"uller geodesic rays to the
boundary at infinity of Teichm\"uller space, an effective estimate comparing
the Teichm\"uller and Thurston metrics along mapping class group orbits of
Teichm\"uller space, and, in the sequel, effective estimates for countings of
filling closed geodesics on closed, negatively curved surfaces.
