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

TL;DR

This paper proves that any automorphism of the Turing degrees induced by a measure-preserving homeomorphism of the Cantor space must be trivial, showing permutations of natural numbers induce only trivial automorphisms.

Contribution

It establishes that permutations of the natural numbers induce only trivial automorphisms of the Turing degrees, under measure-preserving conditions.

Findings

Any automorphism induced by a measure-preserving homeomorphism is trivial.

Permutations of the natural numbers induce only trivial automorphisms.

Automorphisms induced by permutations are trivial.

Abstract

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.