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**1 - 2**of**2**### Extensions, Automorphisms, and Definability

- CONTEMPORARY MATHEMATICS

"... This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to ..."

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This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here

### Orbits of Computably Enumerable Sets: Avoiding an Upper Cone

, 2007

"... 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 ..."

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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. 1