Tel-Aviv University
School of Mathematical Sciences

Department Colloquium

Monday, April 4, 2016

Schreiber 006, 12:15

Zemer Kosloff


Recurrent Anosov Diffeomorphisms without an absolutely continuous invariant measure

By a result of Sinai and Livsic (1972) a generic C2 Anosov diffeomorphism of a compact Riemannian manifold without boundary has no volume (Lebsegue) absolutely continuous invariant measure. On the other hand, Gurevic and Oseldec have shown that every recurrent C2 Anosov diffeomorphism is ergodic and there exists an invariant measure with a smooth density.
In this talk I will explain these notions and discuss my construction of C1 conservative (satisfy Poincare's recurrence) Anosov Diffeomorphism of 𝕋2 without a Lebesgue absolutely continuous invariant measure. Moreover these transformations are of stable (Krieger) type III1 which means that for any ergodic probability preserving transformation (Y,𝒞,ν,S), the Cartesian square f S is a type III1 transformation of (𝕋2 Y,Leb𝕋2 ν). This shows that the class of C1 Anosov diffeomorphisms is dynamically more rich than it's C1+α counterpart.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006