A set
A
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
E
\mathcal {E}
denote the structure of the computably enumerable sets under inclusion,
E
=
(
{
W
e
}
e
∈
ω
,
⊆
)
\mathcal {E} = ( \{ W_e \}_{e\in \omega }, \subseteq )
. Most previously known automorphisms
Φ
\Phi
of the structure
E
\mathcal {E}
of sets were effective (computable) in the sense that
Φ
\Phi
has an effective presentation. We introduce here a new method for generating noneffective automorphisms whose presentation is
Δ
3
0
\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
E
\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
n>0
the well-known degree classes
L
n
\mathbf {L}_n
(the low
n
_n
c.e. degrees) and
H
¯
n
=
R
−
H
n
\overline {\mathbf {H}}_n = \mathbf {R} - \mathbf {H}_n
(the complement of the high
n
_n
c.e. degrees) are noninvariant classes.