The $\Delta _3^0$-automorphism method and noninvariant classes of degrees

A set $A$ of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let $\mathcal {E}$ denote the structure of the computably enumerable sets under inclusion, $\mathcal {E} = ( \{ W_e \}_{e\in \omega }, \subseteq )$. Most previously known automorphisms $\Phi$ of the structure $\mathcal {E}$ of sets were effective (computable) in the sense that $\Phi$ has an effective presentation. We introduce here a new method for generating noneffective automorphisms whose presentation is $\Delta ^0_3$, and we apply the method to answer a number of long open questions about the orbits of c.e. sets under automorphisms of $\mathcal {E}$. For example, we show that the orbit of every noncomputable ( i.e., nonrecursive) c.e. set contains a set of high degree, and hence that for all $n>0$ the well-known degree classes $\mathbf {L}_n$ (the low$_n$ c.e. degrees) and $\overline {\mathbf {H}}_n = \mathbf {R} - \mathbf {H}_n$ (the complement of the high$_n$ c.e. degrees) are noninvariant classes.