To any dynamical system equipped with a metric, we associate a class o
f ''well approximate'' sets. In the case of an expanding rational map
of the Riemann sphere acting on its Julia set, we estimate and in some
cases compute the Hausdorff dimension of the associated ''well approx
imable'' sets. The methods used show a clear link between distortion p
roperties and the type of results obtained in this paper, via ergodic
theory and ubiquity.