Martin’s Conjecture for Borel Turing-Invariant Functions

Let \(2^\omega\) be the set of subsets of \(\mathbb N\), identified with the reals, and let \(\le_T\), \(\equiv_T\), and \(x'\) denote Turing reducibility, Turing equivalence, and the Turing jump. A map \(f:2^\omega\to2^\omega\) is Turing-invariant if \(x\equiv_Ty\) implies \(f(x)\equiv_Tf(y)\). A Turing cone is a set \(C_z=\{x:z\le_Tx\}\). For Borel Turing-invariant maps define
\[
  f\le_M g
  \quad\Longleftrightarrow\quad
  (\exists z)(\forall x\in C_z)\ f(x)\le_Tg(x),
\]
and write \(f\equiv_Mg\) when both \(f\le_Mg\) and \(g\le_Mf\). Then:

1. Every Borel Turing-invariant \(f\) is either constant in Turing degree on a cone, or \(\operatorname{id}\le_Mf\).

2. The quotient by \(\equiv_M\) of the maps satisfying \(\operatorname{id}\le_Mf\) is well-ordered by \(\le_M\), and the immediate successor of \([f]\) is represented by \(x\mapsto f(x)'\).