I presented how surface diffeomorphisms are Borel conjugate to Markov shifts up to zero entropy measures (joint work with Mike Boyle). I stressed the soft part of the result that applies to higher dimensional diffeomorphisms with hyperbolic measures but neglects the measures maximizing the entropy at their period.

The spectral decomposition recently obtained with Sylvain Crovisier and Omri Sarig shows that the C infinity surface diffeos are completely classified by their measures maximizing the entropy at their periods.

(revised slides)

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