American Institute of Mathematical Sciences

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

Smoothness of solenoidal attractors

 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)$f$. 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  [1] Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101 [2] Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35 [3] Patrick Foulon, Boris Hasselblatt. Lipschitz continuous invariant forms for algebraic Anosov systems. Journal of Modern Dynamics, 2010, 4 (3) : 571-584. doi: 10.3934/jmd.2010.4.571 [4] Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002 [5] Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise$C^2$and expanding mappings in higher dimensions. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451 [6] Adrian Tudorascu. On absolutely continuous curves of probabilities on the line. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5105-5124. doi: 10.3934/dcds.2019207 [7] Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80 [8] Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013 [9] Petr Kůrka. On the measure attractor of a cellular automaton. Conference Publications, 2005, 2005 (Special) : 524-535. doi: 10.3934/proc.2005.2005.524 [10] Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365 [11] Rafael De La Llave, Victoria Sadovskaya. On the regularity of integrable conformal structures invariant under Anosov systems. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 377-385. doi: 10.3934/dcds.2005.12.377 [12] Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223 [13] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [14] Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811 [15] Anja Randecker, Giulio Tiozzo. Cusp excursion in hyperbolic manifolds and singularity of harmonic measure. Journal of Modern Dynamics, 2021, 17: 183-211. doi: 10.3934/jmd.2021006 [16] S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593 [17] Delin Wu and Chengkui Zhong. Estimates on the dimension of an attractor for a nonclassical hyperbolic equation. Electronic Research Announcements, 2006, 12: 63-70. [18] Pengfei Zhang. Partially hyperbolic sets with positive measure and$ACIP\$ for partially hyperbolic systems. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435 [19] Hieu Trung Do, Thomas A. Schmidt. New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant. Journal of Modern Dynamics, 2016, 10: 541-561. doi: 10.3934/jmd.2016.10.541 [20] Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

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