Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps

Let $f:[0,1]\to [0,1]$ be a piecewise expanding unimodal map of class $C^{k+1}$, with $k\ge 1$, 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.