Extensive escape rate in lattices of weakly coupled expanding maps with holes

TL;DR

This paper investigates the escape rate in infinite lattices of weakly coupled expanding maps, proposing normalized escape rate as a key measure, supported by symbolic dynamics and a diffusive example showing decay with coupling.

Contribution

It introduces the normalized escape rate for infinite coupled maps and demonstrates its linear growth in periodic approximations, using symbolic dynamics techniques.

Findings

Normalized escape rate grows linearly with period size

Escape rate decreases monotonically with coupling strength

Symbolic dynamics effectively controls perturbation effects

Abstract

This paper discusses possible approaches to the escape rate in infinite lattices of weakly coupled maps with uniformly expanding repeller. It is proved that computed-via-volume rates of spatially periodic approximations grow linearly with the period size, suggesting normalized escape rate as the appropriate notion for the infinite system. The proof relies on symbolic dynamics and is based on the control of cumulative effects of perturbations within cylinder sets. A piecewise affine diffusive example is presented that exhibits monotonic decay of the escape rate with coupling intensity.