# A global structure theorem for measure preserving transformations

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*schedule*
le mardi 21 mars 2017 de 10h30 à 12h00

**Organisé par :**D. Burguet, P-A. Guihéneuf

**Intervenant :**Matthew Foreman (Matthew Foreman)

**Lieu :**Salle Paul Lévy, Campus Jussieu (salle 113, Tour 16/26)

**Sujet :**A global structure theorem for measure preserving transformations

**Résumé :**

Most structure theorems for measure preserving transformations consider them in isolation, analyzing them one at a time. In this talk I discuss a joint theorem with B. Weiss that shows that two large “ecosystems" of measure preserving transformations are isomorphic. The first consists of all finite entropy transformations with a non-trivial odometer factor. The second consists of all transformations that can be realized as diffeomorphisms of the 2-torus by the (untwisted) Anosov-Katok method of Approximation by Conjugacy. We make these into categories by taking factor maps as morphisms and show that the two categories are isomorphic.

As a consequence we solve (negatively) von Neumann’s 1932 proposal to classify ergodic measure preserving transformations up to conjugacy. Moreover the global structure theorem gives a systematic machine for generating Anosov-Katok diffeomorphisms. Examples built by the machine include ergodic Lebesgue measure preserving diffeomorphisms of the 2-torus with arbitrary Choquet simplexes of measures, rank one diffeomorphisms of the torus, diffeomorphisms of the 2-torus of arbitrary distal height and so on.