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. Examples include a dichotomy for periodic and non-periodic points, Cantor sets on the interval, and submanifolds of Anosov diffeomorphisms on surfaces.