Permutations of the integers induce only the trivial automorphism of the Turing degrees

Abstract

Let $\pi$ be an automorphism of the Turing degrees induces by a homeomorphism $\phi$ of the Cantor space $2^{\omega }$ such that $\phi$ preserves all Bernoulli measures. It is proved that $\pi$ must be trivial. In particular, a permutation of $\omega$ can only induce the trivial automorphism of the Turing degrees.