  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.
