The Delta(0)(3)-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 epsilon denote the structure of the computably enumerable sets under inclusion, epsilon = ({W-e}(e is an element of omega), subset of or equal to). Most previously known automorphisms Phi of the structure epsilon 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(3)(0), and we apply the method to answer a number of long open questions about the orbits of c.e. sets under automorphisms of epsilon. 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 L(n) (the low(n) c.e. degrees) and <(H)over bar (n)> = R - H-n (the complement of the high(n) c.e. degrees) are noninvariant classes.