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These lecture notes focus on a recent result of Mike Hochman: an arbitrarystandard Borel system can be embedded into a mixing Markov with equal entropy,respecting all invariant probability measures, with two exceptions: thosecarried by periodic orbits and those with maximal entropy. We discuss thecorresponding notions of almost Borel embedding and isomorphism anduniversality. The main part of this paper is devoted to a self-contained anddetailed proof of Hochman's theorem. We then explain how Katok's horseshoetheorem can be used to analyze diffeomorphisms with "enough" measures that arehyperbolic in the sense of Pesin theory, in both mixing and non-mixingsituations. In the latter setting, new invariants generalizing the measuresmaximizing the entropy emerge.

The almost Borel structure of diffeomorphisms with some hyperbolicity