Martin's Conjecture in the Enumeration Degrees

TL;DR

This paper investigates Martin's Conjecture within enumeration degrees, revealing a broader spectrum of behaviors and demonstrating that uniformly invariant functions are limited to constant, increasing, or skip operator behaviors locally.

Contribution

It extends the analysis of Martin's Conjecture to enumeration degrees, showing the spectrum of behaviors is wider and that uniformly invariant functions have constrained local behavior.

Findings

Uniformly invariant functions are constant, increasing, or above the skip operator locally.

There exists a definable function in enumeration degrees not equivalent to any uniformly invariant function on any cone.

The behavior spectrum in enumeration degrees is much wider than in Turing degrees.

Abstract

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.