School of Mathematical Sciences
Monday, April 4, 2016
Schreiber 006, 12:15
Recurrent Anosov Diffeomorphisms without an absolutely continuous invariant measure
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 ª S is a type III1 transformation
of (𝕋2 ª Y,Leb𝕋2 ª 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