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Zbl 1092.37017
Bamón, Rodrigo; Kiwi, Jan; Rivera-Letelier, Juan; Urzúa, Richard
On the topology of solenoidal attractors of the cylinder. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 2, 209-236 (2006). ISSN 0294-1449

Summary: We study the dynamics of skew product endomorphisms acting on the cylinder $\Bbb R/\Bbb Z\times\Bbb R$, of the form $$(\theta,t)\mapsto (\ell\theta+ \tau(\theta)),$$ where $\ell\ge 2$ is an integer, $\lambda\in(0,1)$ and $\tau:\Bbb R/\Bbb Z\to\Bbb R$ is a continuous function. We are interested in topological properties of the global attractor $\Omega_{\lambda,\tau}$ of this map. Given $\ell$ and a Lipschitz function $\tau$, we show that the attractor set $\Omega_{\lambda,\tau}$ is homeomorphic to a closed topological annulus for all $\lambda$ sufficiently close to 1. Moreover, we prove that $\Omega_{\lambda,\tau}$ is a Jordan curve for at most finitely many $\lambda\in(0,1)$. These results rely on a detailed study of iterated ``cohomological'' equations of the form $\tau={\cal L}_{\lambda_1}\mu_1$, $\mu_1={\cal L}_{\lambda_2}\mu_2,\dots,$ where ${\cal L}_\lambda\mu= \mu\circ{\bold m}_\ell- \lambda\mu$ and ${\bold m}_\ell:\Bbb R/\Bbb Z\to\Bbb R/\Bbb Z$ denotes the multiplication by $\ell$ map. We show the following finiteness result: each Lipschitz function $\tau$ can be written in a canonical way as, $$\tau={\cal L}_{\lambda_1}\circ\cdots\circ{\cal L}_{\lambda_m}\mu,$$ where $m\ge 0$, $\lambda_1,\dots,\lambda_m\in (0,1)]$, and the Lipschitz function $\mu$ satisfies $\mu\ne{\cal L}_\lambda\rho$ for every continuous function $\rho$ and every $\lambda\in(0,1]$.

MSC 2000:: *37C70 Attractors and repellers, topological structure
37E99 Low-dimensional dynamical systems

Keywords: attractors; endomorphisms

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Zentralblatt MATH Berlin [Germany]