School of Mathematical Sciences

Department Colloquium

Schreiber 006, 12:15

Warwick

By a result of Sinai and Livsic (1972) a generic C

Abstract:^{2}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 C^{2}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 C^{1}conservative (satisfy Poincare's recurrence) Anosov Diffeomorphism of 𝕋^{2}without a Lebesgue absolutely continuous invariant measure. Moreover these transformations are of stable (Krieger) typeIII_{1}which means that for any ergodic probability preserving transformation (Y,𝒞,ν,S), the Cartesian square f ª S is a typeIII_{1}transformation of (𝕋^{2}ª Y,Leb_{𝕋2}ª ν). This shows that the class of C^{1}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