Bi-uniformly E0-invariant homeomorphisms of Cantor space induce only the identity automorphism on the Turing degrees.
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A tractable case of the Turing automorphism problem: bi-uniformly $E_0$-invariant Cantor homeomorphisms
Bi-uniformly E0-invariant homeomorphisms of Cantor space induce only the identity automorphism on the Turing degrees.