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

Quantitative recurrence and the shrinking target problem for overlapping iterated function systems