Smoothness of solenoidal attractors

Artur Avila; Sébastien Gouëzel; Masato Tsujii

doi:10.3934/dcds.2006.15.21

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2006, Volume 15, Issue 1: 21-35. Doi: 10.3934/dcds.2006.15.21

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Smoothness of solenoidal attractors

1.
Laboratoire de Probabilités et Modèles aléatoires, Université Pierre et Marie Curie, Boîte courrier 188,75252, Paris Cedex 05
2.
Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 rue d'Ulm, Paris
3.
Department of Mathematics, Hokkaido University, Kita 10 Nishi 8, Sapporo, 060-0810

Received: March 2005

Revised: November 2005

Published: February 2006

Abstract / Introduction Related Papers Cited by

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,\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) $f$.

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

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