Targets and Holes

TL;DR

This paper investigates how a one-dimensional dynamical system's extreme values are affected when it approaches a target while avoiding a small hole, revealing the dependence of the extremal index on escape rates.

Contribution

It introduces a novel analysis of the extremal index in dynamical systems with holes, linking it explicitly to escape rates and extending extremal value theory.

Findings

Extremal index depends explicitly on escape rate.

The presence of holes influences the system's extreme value statistics.

New theoretical framework for dynamical systems with targets and holes.

Abstract

We address the extreme value problem of a one-dimensional dynamical system approaching a fixed target while constrained to avoid a fixed set which can be thought of as a small hole. The presence of the latter influences the extremal index which will now depend explicitly on the escape rate.