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

Gabriel Fuhrmann; Jing Wang

doi:10.3934/dcds.2017249

Article Contents

2017, Volume 37, Issue 11: 5747-5761. Doi: 10.3934/dcds.2017249

This issue Previous Article A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions Next Article On the uniqueness of an ergodic measure of full dimension for non-conformal repellers

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

Gabriel Fuhrmann^1, and
Jing Wang^2, ,

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
* Corresponding author: Jing Wang

Received: May 31, 2016

Revised: May 31, 2017

Published: October 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 37C45, 37D25, 37C70.

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