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 $r \geq 2$. We prove that if $|(\det C)(\det E)| \|C^{-1}\|^{-2s}>1$ 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 $f$.