Statistics of closest return for some non-uniformly hyperbolic systems

For non-uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for nonuniformly hyperbolic dynamical systems developed by L. S. Young.