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Smoothness of solenoidal attractors

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  • 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,\qquad T(x,y)=(l x, \lambda y+f(x)) \

    where l ≥ 2, $0<\lambda<1$ and $f$ is a $C^{r}$ function on $S^{1}$. We show that, if $\lambda^{1+2s}l>1$ 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) $f$.

    Mathematics Subject Classification: Primary: 37C40, 37C30; Secondary: 37D20.

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