The systems we consider are homeomorphisms of compact metric spaces, shifts of finite type, countable state Markov shifts, surface diffeomorphisms which are C^{1+} (have Holder continuous derivative), and isomorphisms of standard Borel spaces.

The "almost-Borel" viewpoint is to consider a system with respect to all its invariant Borel probabilities, neglecting sets which have measure zero for all of them. We consider issues around the classification of systems in this category, and give complete (or close to complete) results for the Markov shifts and C^{1+} surface diffeomorphisms.

The basis for this comes from Mike Hochman, who showed that various familiar systems have a universality property: all Borel systems of smaller entropy can be embedded into them (modulo those universal null sets). Such a system has a universal part (for this category), supporting its nonatomic ergodic measures of less than maximum entropy, and a top part supporting the measures of maximum entropy. The strictly universal part is uniquely determined up to Borel isomorphism (modulo the universal null sets). Thus the entropy and the almost-Borel structure of the top part determine the system up to almost-Borel isomorphism. The universal part is fantastically complicated, but it is homogeneous. The idiosyncratic part at the top is sometimes easily classified.

We extend Hochman's results to more general systems. This requires considering universality and entropy relative to periods. From Hochman's previous work and proper definitions, one gets a complete almost-Borel classification of finite entropy countable state Markov shifts (NOT necessarily irreducible).

The main payoff: up to a set (often empty) supporting some zero entropy measures, every C^{1+} surface diffeomorphism is almost Borel isomorphic to a countable state Markov shift. This result rests on the remarkable symbolic dynamics of Omri Sarig, and an analysis of their factors under this symbolic dynamics with respect to measures with entropy "maximal at a period".

One of the basic open questions about generalizing these results arises from the work of Quas and Soo.