  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.
