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Smoothness of solenoidal attractors

Artur Avila·Sebastien Gouezel·Masato Tsujii
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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, λy+f(x)) $ where $\ell\geq 2$, $0<λ<1$ and $f$ is a $C^r$ function on $S^1$. We show that, if $λ^{1+2s}\ell>1$ 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) $f$.

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Artur Avila, Sebastien Gouezel, Masato Tsujii. 2005-03-08. Smoothness of solenoidal attractors. https://arxiv.org/abs/math/0503144

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