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 $$ \tht \mapsto (\ell θ, \la θ+ τ(θ)), $$ where $ \ell \geq 2$ is an integer, $\la \in (0,1)$ and $τ: \T \to \R$ 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 $τ$, 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 $τ= \cL_{\la_1} μ_1$, $μ_1 = \cL_{\la_2} μ_2, >...$, where $\cL_\la μ= μ\circ \ml - \la μ$ and $\ml: \T \to \T$ denotes the multiplication by $\ell$ map. We show the following finiteness result: each Lipschitz function $τ$ can be written in a canonical way as, $$ τ= \cL_{\la_1} \circ ... \circ \cL_{\la_m} μ, $$ where $m \ge 0$, $λ_1, ..., λ_m \in (0, 1]$ and the Lipschitz function $μ$ satisfies $μ\neq \cL_\la ρ$ for every continuous function $ρ$ and every $\la \in (0,1]$.