The Orbit Equivalence Theorem

Abstract

This chapter begins Part 4 of the monograph. The goal of this part is to prove the Orbit Equivalence Theorem and the Quasi-Isomorphism Theorem. Theorem 17.1 (Orbit Equivalence) states that there is a dynamically large subset Z ⊂ X and a map Ω‎: Z → Y. Section 17.2 defines Z. Section 17.3 defines Ω‎. Section 17.4 characterizes the image Ω‎(Z). Section 17.5 defines a partition of Z into small convex polytopes which have the property that all the maps in Equations 17.1 and 1 are entirely defined and projective on each polytope. This allows us to verify the properties in the Orbit Equivalence Theorem just by checking what the two relevant maps do to the vertices of the new partition. Section 17.6 puts everything together and prove the Orbit Equivalence Theorem modulo some integer computer calculations. Section 17.7 discusses the computational techniques used to carry out the calculations from Section 17.6. Section 17.8 explains the calculations.