# The isomorphism class of the shift map

## TL;DR

This paper investigates the complexity of the shift map's isomorphism class on a special topological space, showing it cannot be distinguished from its inverse by simple topological properties, and explores implications under certain set-theoretic axioms.

## Contribution

It proves that the isomorphism classes of the shift map and its inverse cannot be separated by Borel sets, and analyzes the nature of continuous images of the shift map under different set-theoretic assumptions.

## Key findings

- The isomorphism classes of $\sigma$ and $\sigma^{-1}$ are not Borel separable.
- Under $ extsf{OCA}+ extsf{MA}$, the set of continuous images of $\sigma$ is non-Borel.
- The result contrasts with the case under $ extsf{CH}$ where continuous images form a closed set.

## Abstract

The \emph{shift map} $\sigma$ is the self-homeomorphism of $\omega^* = \beta\omega \setminus \omega$ induced by the successor function $n \mapsto n+1$ on $\omega$. We prove that the isomorphism classes of $\sigma$ and $\sigma^{-1}$ cannot be separated by a Borel set in $\mathcal H(\omega^*)$, the space of all self-homeomorphisms of $\omega^*$ equipped with the compact-open topology. Van Douwen proved it is consistent for $\sigma$ and $\sigma^{-1}$ not to be isomorphic. Whether it is also consistent for them to be isomorphic is an open problem. The theorem stated above can be thought of as a counterpoint to van Douwen's result: while $\sigma$ and $\sigma^{-1}$ may not be isomorphic, there is no simple topological property that distinguishes them. As a relatively straightforward consequence of the main theorem, we deduce that $\mathsf{OCA}+\mathsf{MA}$ implies the set of continuous images of $\sigma$ fails to be Borel in $\mathcal H(\omega^*)$. (Here a ``continuous image'' of $\sigma$ is meant in the sense of topological dynamics: any $h \in \mathcal H(\omega^*)$ such that $q \circ \sigma = h \circ q$ for some continuous surjection $q: \omega^* \to \omega^*$.) This contrasts starkly with a recent theorem of the author showing that under $\mathsf{CH}$, the continuous images of $\sigma$ form a closed subset of $\mathcal H(\omega^*)$.