Pseudojumps and $\Pi_1^0$ Classes

For a pseudojump V^X and a $\Pi_1^0$ class P, we consider properties of the set {V^X:X ∈ P}. We show that if P is Medvedev complete or if P has positive measure, and Ø′ ≤T C, then there exists X ∈ P with V^X ≡T C. We examine the consequences when V^X is Turing incomparable with V^Y for X ≠ Y in P and when $W_e^X=W_e^Y$ for all X, Y ∈ P. Finally, we give a characterization of the jump in terms of $\Pi_1^0$ classes.