# American Institute of Mathematical Sciences

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November  2017, 37(11): 5747-5761. doi: 10.3934/dcds.2017249

## Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent

 1 Institute of Mathematics, Friedrich-Schiller-Universität Jena, Jena 07743, Germany 2 Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China

* Corresponding author: Jing Wang

Received  June 2016 Revised  June 2017 Published  July 2017

We study order-preserving $\mathcal{C}^1$-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.

Citation: Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249
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