# Subshifts of Finite Type with a Hole

## TL;DR

This paper investigates how the escape rate in subshifts of finite type with holes relates to the properties of the holes, such as their correlation and minimal period, providing generalizations of previous results for simpler systems.

## Contribution

It introduces a comprehensive analysis of escape rates in subshifts with multiple holes, relating them to correlations and minimal periods, extending prior work on simpler systems.

## Key findings

- Escape rate increases with the correlation function r(z) for large q.
- Larger minimal period of holes leads to faster escape rates when cross-correlation is zero.
- Counterexamples show the relationship fails with non-zero cross-correlation.

## Abstract

This paper examines the relationship between the escape rate and the minimal period of the hole. We consider a subshift of finite type on $q$ symbols with a union of $t$ cylinders based at words of identical length $p$ as the hole. The escape rate into the hole relates to the asymptotic behavior of the number of words of fixed length that do not contain the fixed set of (forbidden) words at which the cylinders are based. We explore the relationship between the escape rate and $r(z)$, a rational function of the correlations between the forbidden words. In particular, we prove that there exists a constant $D(t,p)$ such that if $q>D(t,p)$, then the escape rate is faster for the hole with larger value of $r(z)$ evaluated at $D(t,p)$. Further, we consider two holes each of which is a union of cylinders based at words of equal length, having zero cross-correlation, and prove that the larger is the minimal period of the collection, the faster is the escape rate. However, when the cross-correlations are non-zero, we give examples to prove that this result fails to hold. Our results are more general than the existing ones known for maps conjugate to a full shift with a single cylinder as the hole. The existing results arise as a special case of our results.