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Smoothness of solenoidal attractors. (English) Zbl 1106.37015
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 $$T$$.

##### 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
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