On the Topology of Solenoidal Attractors of the Cylinder

Abstract

We study the dynamics of skew product endomorphisms acting on the cylinder $\cyl$, of the form $$ \tht \mapsto (\ell \theta, \la \theta + \tau (\theta)), $$ where $ \ell \geq 2$ is an integer, $\la \in (0,1)$ and $\tau: \T \to \R$ is a continuous function. We are interested on {\it topological} properties of the global attractor $\Omegalt$ of this map. Given $\ell$ and a Lipschitz function $\tau$, we show that the attractor set $\Omegalt$ is homeomorphic to a closed topological annulus for all $\la$ sufficiently close to 1. Moreover, we prove that $\Omegalt$ is a Jordan curve for at most finitely many $\la \in (0,1)$. These results rely on a detailed study of iterated ``cohomological'' equations of the form $\tau = \cL_{\la_1} \mu_1$, $\mu_1 = \cL_{\la_2} \mu_2, >...$, where $\cL_\la \mu = \mu \circ \ml - \la \mu$ and $\ml: \T \to \T$ denotes the multiplication by $\ell$ map. We show the following finiteness result: each Lipschitz function $\tau$ can be written in a canonical way as, $$ \tau = \cL_{\la_1} \circ ... \circ \cL_{\la_m} \mu, $$ where $m \ge 0$, $\lambda_1, ..., \lambda_m \in (0, 1]$ and the Lipschitz function $\mu$ satisfies $\mu \neq \cL_\la \rho$ for every continuous function $\rho$ and every $\la \in (0,1]$.