Shrinking targets versus recurrence: A survey on the Hausdorff dimension theory

Abstract

This survey reviews results on the Hausdorff dimension of shrinking target sets and recurrence sets in dynamical systems. In their basic formulations, shrinking target sets concern orbits approaching a prescribed centre, whereas recurrence sets concern orbits returning close to their initial points. A central theme is the comparison of the Hausdorff dimensions of these two classes of limsup sets. The dynamical systems considered include expanding conformal systems, irrational rotations, affine iterated function systems, and matrix transformations. Related results on Hausdorff measure are also included when they provide natural refinements of the dimension theory. Finally, we review results on badly approximating orbits and non-recurrent sets, with particular attention to winning properties and intersections with self-conformal and self-affine sets.