arXiv subjects
Publications and source records attributed to Antonio Nakid Cordero.
Motivated by Ziegler's computability-theoretic characterisation of finite absolute presentability between groups, we prove an analogous theorem in symbolic dynamics. We introduce the notion of one subshift being finitely determined over another and show that this relation is characterised by Ziegler reducibility between their co-languages. We further investigate a notion of existential closure for subshifts.
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.