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

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. J. W. was supported by a research fellowship of the Alexander von Humboldt-Foundation.

Rectifiability; a survey

“…Fuhrmann and Wang [FW17] proved that ergodic measures of certain dynamical systems on the 2-torus are 1-rectifiable. 17.3.…”

Classification of linear skew-products of the complex plane and an affine route to fractalization

“…Normal forms and Lyapunov exponents. Let us consider a linear quasiperiodic skew product as defined in (7), given by a ∈ C r (T, C), r ≥ 0. We have shown that the winding number of a is preserved by linear changes of variables (see Proposition 1) so that it can be seen as an invariant of the cocycle.…”

“…parameters is only continuous (and never differentiable) when the winding number changes (see Section 4 for details). The dynamics of general (nonlinear) skew products is a well known topic in dynamical systems that has been considered by several authors (see, for instance, [25,15,8]), and very often specifically to study the existence of invariant curves, the fractalization phenomenon and the existence of Strange Non-chaotic Attractors (SNAs) [22,23,12,11,13,14,16,2,7,6]. In linear systems, there always is an invariant curve given by z = 0.…”