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

TL;DR

This paper investigates the rectifiability of invariant measures for certain circle diffeomorphisms driven by irrational rotations, showing that most measures in specific families are absolutely continuous on Lipschitz graphs.

Contribution

It establishes the existence of open sets of diffeomorphism families where invariant measures are predominantly one-rectifiable, advancing understanding of measure regularity in dynamical systems.

Findings

Most invariant measures are one-rectifiable in the studied families.

Invariant measures are absolutely continuous with respect to Hausdorff measure on Lipschitz graphs.

The results apply to order-preserving C^1-circle diffeomorphisms with Diophantine rotation numbers.

Abstract

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.