Smoothness of solenoidal attractors. (English) Zbl 1106.37015

Avila, Artur; Gouëzel, Sébastien; Tsujii, Masato

Discrete Contin. Dyn. Syst. 15, No. 1, 21-35 (2006).

The authors deal with dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle \(S^1= \mathbb{R}/\mathbb{Z}\): \[ T: S^1\times\mathbb{R}\to S^1\times\mathbb{R},\quad T(x,y)= (\ell x,\lambda y+ f(x)), \] where \(\ell\geq 2\) is an integer, \(\lambda\in (0,1)\) and \(f\in C^r(S^1)\), \(r> 3\). The authors study the smoothness of the density of the SBR measure in more detail, and the mixing properties of \(T\).

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

37C70	Attractors and repellers of smooth dynamical systems and their topological structure
37C40	Smooth ergodic theory, invariant measures for smooth dynamical systems
37C30	Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37A25	Ergodicity, mixing, rates of mixing
37E10	Dynamical systems involving maps of the circle

Keywords:

skew-product; mixing property; SBR measure; maps on the circle