Smoothness of solenoidal attractors

Abstract

We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle $S^1$: $ T:S^{1}\times \R\to S^1\times \R, T(x,y)=(\ell x, \lambda y+f(x)) $ where $\ell\geq 2$, $0<\lambda<1$ and $f$ is a $C^r$ function on $S^1$. We show that, if $\lambda^{1+2s}\ell>1$ for some $0\leq s< r-2$, the density of the SBR measure for $T$ is contained in the Sobolev space $W^s(S^1\times \R)$ for almost all ($C^r$ generic, at least) $f$.