The isomorphism class of the shift map

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 $\mathcal H(\omega^*)$.