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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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]

Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451

[5]

Adrian Tudorascu. On absolutely continuous curves of probabilities on the line. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5105-5124. doi: 10.3934/dcds.2019207

[6]

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

[7]

Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013

[8]

Petr Kůrka. On the measure attractor of a cellular automaton. Conference Publications, 2005, 2005 (Special) : 524-535. doi: 10.3934/proc.2005.2005.524

[9]

Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365

[10]

Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223

[11]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[12]

Rafael De La Llave, Victoria Sadovskaya. On the regularity of integrable conformal structures invariant under Anosov systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 377-385. doi: 10.3934/dcds.2005.12.377

[13]

Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811

[14]

S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593

[15]

Delin Wu and Chengkui Zhong. Estimates on the dimension of an attractor for a nonclassical hyperbolic equation. Electronic Research Announcements, 2006, 12: 63-70.

[16]

Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435

[17]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[18]

Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008

[19]

Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123

[20]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]