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

TL;DR

This paper investigates the relationship between escape rates and extremal indices in dynamical systems, proving convergence under certain mixing conditions and establishing equivalence between entry and return escape rates.

Contribution

It introduces a new connection between local escape rates and extremal indices, and proves convergence results under polynomial $ ext{phi}$-mixing conditions.

Findings

Localized escape rate converges to the extremal index for $ ext{phi}$-mixing systems.

Established equivalence between entry and return local escape rates.

Provides theoretical foundation for understanding escape dynamics in measure-zero sets.

Abstract

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.