Let $\mathcal{D}$ be the collection of Turing degrees. Are there nontrivial maps $\phi:\mathcal{D}\to \mathcal{D}$ which is natural to consider? For instance, I wonder whether maps which are continuous with respect to the Scott topology or the poset topology on $\mathcal{D}$ have been studied.
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asked Feb 22, 2022 at 17:19
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3$\begingroup$ Look up "Martin's conjecture" - very vaguely, it's believed that any reasonably-definable map on the Turing degrees is something like an iterate of the Turing jump. $\endgroup$– Noah SchweberCommented Feb 22, 2022 at 18:16
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