TOPOLOGICAL ORBIT EQUIVALENCE AND FACTOR MAPS IN SYMBOLIC DYNAMICS

Chapter I, "Lower Entropy Factors of Sofic Systems", will appear as an article. The main result: if S and T are irreducible subshifts of finite type, the period of any periodic point of S is divisible by the period of some point of T and the entropy of S is strictly greater than the entropy of T, then T is a factor of S. In Chapter II, homeomorphisms S and T on compact metric spaces are called (topologically) orbit equivalent if some homeomorphism takes orbits of S onto orbits of T. A topological analogue of Belinskaya's theorem is obtained: if S and T are transitive and orbit equivalent by continuous jumps, then S is isomorphic to T or T('-1) by continuous jumps. If S and T are transitive subshifts of finite type orbit equivalent by bounded jumps, then S is isomorphic to T or T('-1); this result fails for sofic shifts. If S and T are orbit equivalent mixing sofic shifts, then S need not be isomorphic to T or T('-1). However, the maximal measures of orbit equivalent mixing sofic shifts must have the same range of values on cylinder sets. Orbit equivalence does not respect expansiveness or specification, and displays other pathology. In Chapter III, an example is given of a sofic shift which is not "almost finite type", in the sense of Marcus.