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February  2006, 15(1): 21-35. doi: 10.3934/dcds.2006.15.21

Smoothness of solenoidal attractors

Artur Avila 1, , Sébastien Gouëzel 2, and  Masato Tsujii 3,

1. 

Laboratoire de Probabilités et Modèles aléatoires, Université Pierre et Marie Curie, Boîte courrier 188,75252, Paris Cedex 05, France
2. 

Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 rue d'Ulm, Paris, France
3. 

Department of Mathematics, Hokkaido University, Kita 10 Nishi 8, Sapporo, 060-0810, Japan

Received  March 2005 Revised  November 2005 Published  February 2006

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) $f$.

Keywords: invariant measure., Hyperbolic attractor, Anosov endomorphism, absolutely continuous.
Mathematics Subject Classification: Primary: 37C40, 37C30; Secondary: 37D2.
Citation: Artur Avila, Sébastien Gouëzel, Masato Tsujii. Smoothness of solenoidal attractors. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 21-35. doi: 10.3934/dcds.2006.15.21
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