Automorphisms of the lattice of Pi(0)(1) classes; perfect thin classes andANC degrees

Pi (0)(1) classes are important to the logical analysis of many parts of ma thematics. The Pi (0)(1) classes form a lattice. As with the lattice of com putably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, o r more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin c lasses. Our main results are an analog of Martin's work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog o f Soare's work demonstrating that maximal sets form an orbit. In particular , we show that the collection of perfect thin classes (a notion which is de finable in the lattice of Pi (0)(1) classes) forms an orbit in the lattice of Pi (0)(1) classes; and a degree is anc iff it contains a perfect thin cl ass. Hence the class of anc degrees is an invariant class for the lattice o f Pi (0)(1) classes. We remark that the automorphism result is proven via a Delta (0)(3) automorphism, and demonstrate that this complexity is necessa ry.