Martin's Conjecture in the Enumeration Degrees

Martin's Conjecture states that every definable function on the Turing degrees is either constant or increasing, and that every increasing function is an iterate of the Turing jump. This classification has already been corroborated for the class of uniformly invariant functions and a long-standing conjecture by Steel is that every definable function on the Turing degrees is equivalent to a uniformly invariant one. We explore whether a similar classification is possible in the enumeration degrees, an extension of the Turing degrees. We show that the spectrum of behavior is much wider in the enumeration degrees, even for uniformly invariant functions. However, our main result is that uniformly invariant functions behave locally as nicely as possible: they are constant, increasing, or above the skip operator. As a consequence, we show that there is a definable function in the enumeration degrees that is not equivalent to a uniformly invariant one on any cone.