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 $F_\sigma$ set under it is again $F_\sigma$. 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 $\alpha$ and $\beta$ with $\alpha\leq\beta<\alpha\cdot 2$, every function between Polish spaces having small transfinite inductive dimension is decomposable into countably many Baire class $\gamma$ functions with $\mathbf{\Delta}^0_{\beta+1}$ domains such that $\gamma+\alpha\leq\beta$ if and only if the preimage of each $\mathbf{\Sigma}^0_{\alpha+1}$ set under that function is $\mathbf{\Sigma}^0_{\beta+1}$, and the transformation of a $\mathbf{\Sigma}^0_{\alpha+1}$ set into the $\mathbf{\Sigma}^0_{\beta+1}$ preimage is continuous.