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 $S^1$: $T:S^1\times \mathbb{R}\to S^1\times \mathbb{R},T(x,y)=(\ell x,\lambda y+f(x))$ where $\ell \ge 2$, $0<\lambda <1$ and $f$ is a $C^r$ function on $S^1$. We show that, if ${\lambda }^{1+2s}\ell >1$ for some $0\le s<r-2$, the density of the SBR measure for $T$ is contained in the Sobolev space $W^s(S^1\times \mathbb{R})$ for almost all ($C^r$ generic, at least) $f$.

17 pages