A tractable case of the Turing automorphism problem: bi-uniformly $E_0$-invariant Cantor homeomorphisms

Abstract

A function $F:2^{\omega }\to 2^{\omega }$ is an $E_0$-isomorphism if for all $x,y\in 2^{\omega }$, we have $x{E}_{0}y⟺f(x)E_{0}f(y)$, where $x{E}_{0}y⟺(\exists a)(\forall n\ge b)x(n)=y(n)$. If such witnesses $a$ for $x{E}_{0}y$ and for $f(x)E_{0}f(y)$ depend on each other but not on $x$, $y$, then $F$ is called bi-uniform. It is shown that a homeomorphism of Cantor space which is a bi-uniform $E_0$-isomorphism can induce only the trivial automorphism of the Turing degrees.

Accepted for publication in: Higher recursion theory and set theory. Lecture Notes Series, Institute of Mathematical Sciences, National University of Singapore, World Scientific Publishing, Hackensack, NJ