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.