The \Delta^0_2 Turing degrees: Automorphisms and Definability

We prove that the \Delta ^0_2 Turing degrees have a finite automorphism base. We apply this result to show that the automorphism group of {\mathcal D}_T(\leq \mathbf {0'}) is countable and that all its members have arithmetic presentations. We prove that every relation on {\mathcal D}_T(\leq \mathbf {0'}) induced by an arithmetically definable degree invariant relation is definable with finitely many \Delta ^0_2 parameters and show that rigidity for {\mathcal D}_T(\leq \mathbf {0'}) is equivalent to its biinterpretability with first order arithmetic.