Decomposing Borel functions using the Shore-Slaman join theorem

Jayne and Rogers proved that every function from an analytic space into a separable metric space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_σ$ set under it is again $F_σ$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers theorem at finite and transfinite levels of the hierarchy of Borel functions: For all countable ordinals $α$ and $β$ with $α\leqβ<α\cdot 2$, every function between Polish spaces having small transfinite inductive dimension is decomposable into countably many Baire class $γ$ functions with $\mathbfΔ^0_{β+1}$ domains such that $γ+α\leqβ$ if and only if the preimage of each $\mathbfΣ^0_{α+1}$ set under that function is $\mathbfΣ^0_{β+1}$, and the transformation of a $\mathbfΣ^0_{α+1}$ set into the $\mathbfΣ^0_{β+1}$ preimage is continuous.

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