Is there a nontrivial automorphism of the Turing degrees? It is a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be. Here we consider instead how an automorphism might be induced by a function on reals, or even by a function on integers.

We show that a permutation of $\omega$ cannot induce any nontrivial automorphism of the Turing degrees of members of $2^\omega$, and in fact any permutation that induces the trivial automorphism must be computable.

A main idea of the proof is to consider the members of $2^\omega$ to be probabilities, and use statistics: from random outcomes from a distribution we can compute that distribution, but not much more.