A bounded jump for the bounded Turing degrees

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We define the bounded jump of A by Aᵇ = x | Exists i <= x [phiᵢ (x) converges and Phiₓ[A|phiᵢ(x)](x) converges and let A[nb] denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly increasing and order preserving on the bounded Turing (bT) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for n > 1 we have X <=[bT] 0[nb] iff X is omegaⁿ-c.e. iff X <=₁ 0[nb], extending the classical result that X <=[bT] 0' iff X is omega-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every X such that 0ᵇ <=[bT] X <=[bT] 0[2b], there is a Y <=[bT] 0ᵇ such that Yᵇ =[bT] X.