We show that if A and Â are automorphic via Φ then the structures 𝒮ℛ(A) and 𝒮ℛ(Â) are δ03-isomorphic via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.
"Isomorphisms of splits of computably enumerable sets." J. Symbolic Logic 68 (3) 1044 - 1064, September 2003. https://doi.org/10.2178/jsl/1058448453