Faster than expected escape for a class of fully chaotic maps

TL;DR

This paper derives an exact formula for escape rates in hyperbolic systems with Markov holes, revealing that escape can be faster than expected and analyzing how escape rates depend on hole size and position.

Contribution

It introduces a novel periodic orbit formula for finite Markov holes and explores the dependence of escape rates on hole placement and size in chaotic maps.

Findings

Escape rate exceeds expectation for systems conjugate to binary shift.

Difference in escape rates decays quadratically with hole size.

Results suggest potential extensions to non-Markov holes and applications in network dynamics.

Abstract

We investigate the dependence of the escape rate on the position of a hole placed in uniformly hyperbolic systems admitting a finite Markov partition. We derive an exact periodic orbit formula for finite size Markov holes which differs from other periodic expansions in the literature and can account for additional distortion to maps with piecewise constant expansion rate. Using asymptotic expansions in powers of hole size we show that for systems conjugate to the binary shift, the average escape rate is always larger than the expectation based on the hole size. Moreover, we show that in the small hole limit the difference between the two decays like a known constant times the square of the hole size. Finally, we relate this problem to the random choice of hole positions and we discuss possible extensions of our results to non-Markov holes as well as applications to leaky dynamical networks.