Regularity of the Density of SRB Measures for Solenoidal Attractors

Abstract

We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose density are regular. The maps we consider are given by $T(x,y)=(E(x),C(y)+f(x))$, where $E$ is a linear expanding map of $\mathbb{T}$, $C$ is a linear contracting map of ${\mathbb{R}}^d$, $f$ is in $C^r({\mathbb{T}}^u,{\mathbb{R}}^d)$ and $r\ge 2$. We prove that if $|(\det C)(\det E)|\Vert C^{-1}{\Vert }^{-2s}>1$ for some $s<r-(\frac{u+d}{2}+1)$ and $T$ satisfies a certain transversality condition, then the density of the SRB measure of $T$ is contained in the Sobolev space $H^s({\mathbb{T}}^u\times {\mathbb{R}}^d)$, in particular, if $s>\frac{u+d}{2}$ then the density is $C^k$ for every $k<s-\frac{u+d}{2}$. We also exhibit a condition involving $E$ and $C$ under which this tranversality condition is valid for almost every $f$.