Where to place a hole to achieve a maximal escape rate

TL;DR

This paper investigates how the placement of a hole in a chaotic dynamical system affects the escape rate, revealing that the position, especially near points with maximal minimal period, significantly influences escape probabilities.

Contribution

It introduces a novel analysis of how the position of a hole in chaotic maps impacts escape rates, emphasizing the importance of local dynamical features over hole size.

Findings

Escape rate depends on the minimal period at the hole's location.

Maximal escape occurs at the hole with the highest minimal period.

Results hold for all finite times, not just asymptotically.

Abstract

A natural question of how the survival probability depends upon a position of a hole was seemingly never addressed in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period) which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole.