Escape rate and conditional escape rate from a probabilistic point of view

We prove that for a sequence of nested sets \(\{U_n\}\) with \(\Lambda = \cap_n U_n\) a measure zero set, the localized escape rate converges to the extremal index of \(\Lambda\), provided that the dynamical system is \(\phi\)-mixing at polynomial speed. We also establish the general equivalence between the local escape rate for entry times and the local escape rate for returns.