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

General Topology;Logic