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- W4288365572 abstract "The emph{shift map} $sigma$ is the self-homeomorphism of $omega^* = betaomega setminus omega$ 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 $mathcal H(omega^*)$." @default.
- W4288365572 created "2022-07-29" @default.
- W4288365572 creator A5014159083 @default.
- W4288365572 date "2019-04-22" @default.
- W4288365572 modified "2023-09-29" @default.
- W4288365572 title "The isomorphism class of the shift map" @default.
- W4288365572 doi "https://doi.org/10.48550/arxiv.1904.09907" @default.
- W4288365572 hasPublicationYear "2019" @default.
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