Cover times in dynamical systems

What is the expected number of iterates of a point needed for a plot of these iterates to approximate the attractor of the dynamical system up to a given scale delta (i.e., the orbit will have visited a delta-neighbourhood of every point in the attractor)?  This question has analogues in random walks on graphs and Markov chains and can be seen as a recurrence problem.  I'll present joint work with Natalia Jurga (St Andrews) where we estimate the expectation for this problem as a function of delta for some classes of interval maps using ideas from Hitting Time Statistics and transfer operators as well as permutations and inducing.