Abstract: 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,Leb_𝕋² × ν). This shows that the class of C¹ Anosov diffeomorphisms is dynamically more rich than it's C^1+α counterpart.

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