Rectifiability of a class of invariant measures with one non-vanishing   Lyapunov exponent
  We study order-preserving C^1-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures of nearly all family members are one-rectifiable, that is, absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs. 
Gabriel Fuhrmann, Jing Wang
math.DS