# Entropy pair realization

## TL;DR

This paper demonstrates that for any countable ordinal, there exists a subshift whose entropy pair relation's process of alternating topological and transitive closure terminates precisely at that ordinal, extending the understanding of entropy pair relations.

## Contribution

It constructs subshifts that realize arbitrary countable ordinals as the length of the entropy pair process, combining three novel constructions to achieve this generality.

## Key findings

- Every countable ordinal can be realized as the length of an entropy pair process in a subshift.
- The paper provides a method to implement any abstract process on a zero-dimensional space as a process on a subshift.
- It shows how to realize any shift-invariant relation as the entropy pair relation of a supershift.

## Abstract

We show that the CPE class $\alpha$ of Barbieri and Garc\'ia-Ramos contains a one-dimensional subshift for all countable ordinals $\alpha$, i.e.\ the process of alternating topological and transitive closure on the entropy pairs relation of a subshift can end on an arbitrary ordinal. This is the composition of three constructions: We first realize every ordinal as the length of an abstract "close-up" process on a countable compact space. Next, we realize any abstract process on a compact zero-dimensional metrizable space as the process started from a shift-invariant relation on a subshift, the crucial construction being the implementation of every compact metrizable zero-dimensional space as an open invariant quotient of a subshift. Finally we realize any shift-invariant relation $E$ on a subshift $X$ as the entropy pair relation of a supershift $Y \supset X$, and under strong technical assumptions we can make the CPE process on $Y$ end on the same ordinal as the close-up process of~$E$.