On homeomorphisms of Cantor space that induce only the trivial Turing automorphism
To determine whether there is a nontrivial automorphism of the Turing degrees remains a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be, and ruled out that an automorphism might be induced by a function on integers. 
A homeomorphism $F$ of the Cantor space is said to induce a uniform map mod finite if for each $a\in\omega$ there is a $b\in\omega$ such that for all $X$ and $Y$, if $X(n)=Y(n)$ for all $n\ge b$ then $F(X)(m)=F(Y)(m)$ for all $m\ge a$. 
We show that if such an $F$ induces an automorphism $\pi$ of the Turing degrees, then $\pi$ is trivial. This generalizes, and provides an easier proof of, the past result where $F$ was assumed to be induced by a permutation of $\omega$.