The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors

Extending work of Hochman, we study the almost-Borel structure, i.e., the nonatomic invariant probability measures, of symbolic systems and surface diffeomorphisms. We first classify Markov shifts and characterize them as strictly universal with respect to a natural family of classes of Borel systems. We then study their continuous factors showing that a low entropy part is almost-Borel isomorphic to a Markov shift but that the remaining part is much more diverse, even for finite-to-one factors. However, we exhibit a new condition which we call ‘Bowen type’ which gives complete control of those factors. This last result applies to and was motivated by the symbolic covers of Sarig. We find complete numeric invariants for Borel isomorphism of C1+ surface diffeomorphisms modulo zero entropy measures; for those admitting a totally ergodic measure of positive (not necessarily maximal) entropy, we get a classification up to almostBorel isomorphism.