THE ERGODIC-THEORY OF SHRINKING TARGETS

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.