  We discuss selected topics of current research interest in the theory of
dynamical systems, with emphasis on dimension theory, multifractal analysis,
and quantitative recurrence. The topics include the quantitative versus the
qualitative behavior of Poincar\'e recurrence, the product structure of
invariant measures and return times, the dimension of invariant sets and
invariant measures, the complexity of the level sets of local quantities from
the point of view of Hausdorff dimension, and the conditional variational
principles as well as their applications to problems in number theory.
  We present the foundations of each area, and discuss recent developments and
applications. All the necessary notions from ergodic theory, hyperbolic
dynamics, dimension theory, and the thermodynamic formalism are briefly
recalled. We concentrate on uniformly hyperbolic dynamics, although we also
refer to nonuniformly hyperbolic dynamics. Instead of always presenting the
most general results, we made a selection with the purpose of illustrating the
main ideas while we avoid the accessory technicalities.
