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

中文翻译：

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移位映射的同构类

\emph{shift map} $\sigma$ 是 $\omega^* = \beta\omega \setminus \omega$ 的自同胚，由 $\omega$ 上的后继函数 $n \mapsto n+1$ 诱导. 我们证明了 $\sigma$ 和 $\sigma^{-1}$ 的同构类不能被 $\mathcal H(\omega^*)$ 中的 Borel 集分隔，$\mathcal H(\omega^*)$ 是 $ 的所有自同胚空间\omega^*$ 配备紧凑开放拓扑。Van Douwen 证明 $\sigma$ 和 $\sigma^{-1}$ 不同构是一致的。它们同构是否也一致是一个悬而未决的问题。上述定理可以被认为是范杜文结果的对立面：虽然 $\sigma$ 和 $\sigma^{-1}$ 可能不是同构的，但没有简单的拓扑性质来区分它们。作为主定理的一个相对直接的结果，我们推断 $\mathsf{OCA}+\mathsf{MA}$ 意味着 $\sigma$ 的连续图像集在 $\mathcal H(\omega^*)$ 中不属于 Borel。（这里 $\sigma$ 的“连续图像”在拓扑动力学意义上是指：任何 $h \in \mathcal H(\omega^*)$ 使得 $q \circ \sigma = h \circ q$ 对于一些连续的投影 $q: \omega^* \to \omega^*$.) 这与作者最近的定理形成鲜明对比，该定理表明在 $\mathsf{CH}$ 下，$\sigma 的连续图像$ 形成 $\mathcal H(\omega^*)$ 的封闭子集。