Product of Expansive Markov Maps with Hole

TL;DR

This paper studies how the escape rate in expansive Markov maps with holes depends on the hole's position, revealing that it is not solely determined by the hole's size, and explores related combinatorial problems.

Contribution

It demonstrates the dependence of escape rates on hole position in Markov maps and compares different hole configurations with the same measure.

Findings

Escape rate varies with hole position.

Connected and union of holes with same measure have different escape rates.

Illustrates combinatorial problems related to hole configurations.

Abstract

We consider product of expansive Markov maps on an interval with hole which is conjugate to a subshift of finite type. For certain class of maps, it is known that the escape rate into a given hole does not just depend on its size but also on its position in the state space. We illustrate this phenomenon for maps considered here. We compare the escape rate into a connected hole and a hole which is a union of holes with a certain property, but have same measure. This gives rise to some interesting combinatorial problems.