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We study in this paper estimates on the size of the sets of points which are well approximated by orbits of other points under certain dynamical systems. We apply the results obtained to the particular case of the dynamical system generated by inner functions in the unit disk of the complex plane.

We study the (metric) Diophantine approximation properties of uniformly expanding transformations and some non-uniformly expanding transformations, i.e. transformations T (x) with an associated countable (not necessarily finite) partition and a return time function R(x) (constant on the blocks of the partition) so that T (x) = T R(x) (x) is uniformly expanding, and we obtain Borel-Cantelli results on hitting times of shrinking targets. Our arguments do not require the so-called big image property for T and our results contain most of the diversity of examples of slowly mixing systems. We also obtain, with related techniques, results for one-sided topological Markov chains over a countable alphabet with a Gibbs measure.

We study the set of bounded geodesies of hyperbolic manifolds. For general Riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its Hausdorff dimension. Some applications to diophantine approximation are included.