Why Escape Is Faster Than Expected

TL;DR

This paper investigates escape rates in chaotic dynamical systems with holes, revealing why escape occurs faster than anticipated due to the convexity of the escape rate function, supported by exact calculations.

Contribution

It introduces a new explanation for the unexpectedly rapid escape rates in hyperbolic systems, supported by exact computations for specific maps.

Findings

Escape rate is faster than expected due to convexity.

Exact calculations for skewed tent map and Arnold's cat map.

Comparison of various escape rate estimates.

Abstract

We consider chaotic (hyperbolic) dynamical systems which have a generating Markov partition. Then, open dynamical systems are built by making one element of a Markov partition a hole through which orbits escape. We compare various estimates of the escape rate which correspond to a physical picture of leaking in the entire phase space. Moreover, we uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Exact computations are present for the skewed tent map and Arnold's cat map.