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.

中文翻译：

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从概率的角度看逃逸率和条件逃逸率

我们证明对于一系列嵌套集合\(\{U_n\}\)和\(\Lambda = \cap _n U_n\)一个测度零集，局部逃逸率收敛到\(\Lambda \ )，前提是动态系统是\(\phi \) -以多项式速度混合。我们还建立了进入时间的局部逃逸率和返回的局部逃逸率之间的一般等价性。示例包括对周期性和非周期性点的二分法，区间上的Cantor集以及表面上Anosov微分同形的子流形。