The uniform Martin’s conjecture for many-one degrees
Takayuki Kihara, Antonio Montalbán · Transactions of the American Mathematical Society · 2018
We study functions from reals to reals which are uniformly degree invariant from Turing equivalence to many-one equivalence, and we compare them “on a cone”. We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin’s conjecture for uniformly invariant functions from Turing equivalence to Turing equivalence. Our proof works in the general case of many-one degrees on $\mathcal {Q}^{\omega }$ and Wadge degrees of functions ${\omega }^{\omega }\to \mathcal {Q}$ for any better-quasi-ordering $\mathcal {Q}$.