Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps

TL;DR

This paper investigates the regularity properties of the invariant density for piecewise expanding unimodal maps, showing that non-differentiability points are rare in a measure-theoretic sense and exploring higher-order differentiability.

Contribution

It improves previous results by Szewc (1984) on the measure-theoretic size of non-differentiability points and extends the analysis to higher-order differentiability in the Whitney sense.

Findings

Non-differentiability points have zero Hausdorff dimension.

The set of non-differentiability points can be uncountable if the critical orbit is dense.

Results on higher-order differentiability of the invariant density.

Abstract

Let $f : [0, 1] \to [0, 1]$ be a piecewise expanding unimodal map of class $C^{k+1}$, with $k \geq 1$, and $\mu = \rho dx$ the (unique) SRB measure associated to it. We study the regularity of $\rho$. In particular, points $\mathcal{N}$ where $\rho$ is not differentiable has zero Hausdorff dimension, but is uncountable if the critical orbit of $f$ is dense. This improves on a work of Szewc (1984). We also obtain results about higher orders of differentiability of $\rho$ in the sense of Whitney.