Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure
Abstract
We construct examples of $C^{1}$ Anosov diffeomorphisms on $\mathbb{T}^{2}$ which are of Krieger type ${\rm III}_{1}$ with respect to Lebesgue measure. This shows that the Gurevic Oseledec phenomena that conservative $C^{1+\alpha}$ Anosov diffeomorphisms have a smooth invariant measure does not hold true in the $C^{1}$ setting.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.7707
 Bibcode:
 2014arXiv1410.7707K
 Keywords:

 Mathematics  Dynamical Systems;
 37D20;
 37C40;
 37A40
 EPrint:
 59 pages