We address the extreme value problem of a one-dimensional dynamical system approaching a fixed target while constrained to avoid a fixed set which can be thought of as a small hole. The presence of the latter influences the extremal index which will now depend explicitly on the escape rate.
@article{arxiv.1909.09358,
title = {Targets and Holes},
author = {P. Giulietti and P. Koltai and S. Vaienti},
journal= {arXiv preprint arXiv:1909.09358},
year = {2019}
}