Maximal escape rate for shifts

TL;DR

This paper characterizes the holes with maximal escape rate in shift spaces, revealing complex relationships between escape rate, measure, and periodic orbits, and extends the analysis to Markov measures.

Contribution

It provides a complete characterization of maximal escape rate holes in shift spaces and explores their properties under different measures.

Findings

Holes with maximal escape rate are characterized explicitly.

Ordering holes by escape rate differs from ordering by measure or periodic orbit length.

Preliminary results indicate more complex behavior under Markov measures.

Abstract

We consider the shift transformation on the space of infinite sequences over a finite alphabet endowed with the invariant product measure, and examine the presence of a \emph{hole} on the space. The holes we study are specified by the sequences that do not contain a given finite word as initial sub-string. The measure of the set of sequences that do not fall into the hole in the first $n$ iterates of the shift is known to decay exponentially with $n$, and its exponential rate is called \emph{escape rate}. In this paper we provide a complete characterization of the holes with maximal escape rate. In particular we show that, contrary to the case of equiprobable symbols, ordering the holes by their escape rate corresponds to neither the order by their measure nor by the length of the shortest periodic orbit they contain. Finally, we adapt our technique to the case of shifts endowed with Markov measures, where preliminary results show that a more intricate situation is to be expected.