Orbits of Computably Enumerable Sets: Avoiding an Upper Cone

We investigate the orbit of a low computably enumerable set un-der automorphisms of the partial order E of c.e. sets under inclusion. Given an arbitrary low c.e. set A and an arbitrary noncomputable c.e. set C, we use the New Extension Theorem of Soare to construct an automorphism of E mapping A to a set B such that C 6≤T B. Thus, the orbit in E of the low set A cannot be contained in the upper cone above C. This complements a result of Harrington, who showed that the orbit of a noncomputable c.e. set cannot be contained in the lower cone below any incomplete c.e. set.