Permutations of the integers induce only the trivial automorphism of the   Turing degrees
  Let $\pi$ be an automorphism of the Turing degrees induces by a homeomorphism $\varphi$ of the Cantor space $2^\omega$ such that $\varphi$ 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. 
Bj{\o}rn Kjos-Hanssen
math.LO