The isomorphism class of the shift map

Abstract

The shift map is the self-homeomorphism of ω * = βω \ ω induced by the successor function n → n + 1 on ω. We prove that the isomorphism classes of σ and σ −1 cannot be separated by a Borel set in H(ω * ), the space of all self-homeomorphisms of ω * equipped with the compact-open topology.Van Douwen proved it is consistent for σ and σ −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 σ and σ −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 OCA + MA implies the set of continuous images of σ fails to be Borel in H(ω * ). (Here a "continuous image" of σ is meant in the sense of topological dynamics: any h ∈ H(ω * ) such that q • σ = h • q for some continuous surjection q : ω * → ω * .) This contrasts starkly with a recent theorem of the author showing that under CH, the continuous images of σ form a closed subset of H(ω * ).