Open circle maps: Small hole asymptotics

TL;DR

This paper analyzes escape rates in chaotic maps with small holes, extending known linear results to include higher-order terms and fractal effects, with implications for understanding dynamical systems' sensitivity to phase space perturbations.

Contribution

It introduces a refined asymptotic expansion for escape rates in the doubling map, incorporating a smooth h^2 ln h term and explicit fractal contributions, advancing the understanding of small hole effects.

Findings

Escape rate expansion includes a h^2 ln h term.

Numerical simulations confirm the asymptotic form.

Dependence on hole location involves a dynamical Diophantine condition.

Abstract

We consider escape from chaotic maps through a subset of phase space, the hole. Escape rates are known to be locally constant functions of the hole position and size. In spite of this, for the doubling map we can extend the current best result for small holes, a linear dependence on hole size h, to include a smooth h^2 ln h term and explicit fractal terms to h^2 and higher orders, confirmed by numerical simulations. For more general hole locations the asymptotic form depends on a dynamical Diophantine condition using periodic orbits ordered by stability.