  In this paper we study quantitative recurrence and the shrinking target
problem for dynamical systems coming from overlapping iterated function
systems. Such iterated function systems have the important property that a
point often has several distinct choices of forward orbit. As is demonstrated
in this paper, this non-uniqueness leads to different behaviour to that
observed in the traditional setting where every point has a unique forward
orbit.
  We prove several almost sure results on the Lebesgue measure of the set of
points satisfying a given recurrence rate, and on the Lebesgue measure of the
set of points returning to a shrinking target infinitely often. In certain
cases, when the Lebesgue measure is zero, we also obtain Hausdorff dimension
bounds. One interesting aspect of our approach is that it allows us to handle
targets that are not simply balls, but may have a more exotic geometry.
