The ergodic theory of shrinking targets

SummaryTo any dynamical system equipped with a metric, we associate a class of “well approximable” 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 approximable” sets. The methods used show a clear link between distortion properties and the type of results obtained in this paper, via ergodic theory and ubiquity.

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