On the Topology of Solenoidal Attractors of the Cylinder

We study the dynamics of skew product endomorphisms acting on the cylinder $\text{cyl}$, of the form $$\text{tht}\mapsto (\mathrm{\ell }\theta ,\text{la}\theta +\tau (\theta )),$$ where $\mathrm{\ell }\ge 2$ is an integer, $\text{la}\in (0,1)$ and $\tau :\text{T}\to \mathbb{R}$ is a continuous function. We are interested on {\it topological} properties of the global attractor $\text{Omegalt}$ of this map. Given $\mathrm{\ell }$ and a Lipschitz function $\tau$, we show that the attractor set $\text{Omegalt}$ is homeomorphic to a closed topological annulus for all $\text{la}$ sufficiently close to 1. Moreover, we prove that $\text{Omegalt}$ is a Jordan curve for at most finitely many $\text{la}\in (0,1)$. These results rely on a detailed study of iterated ``cohomological’’ equations of the form $\tau ={\text{cL}}_{{\text{la}}_1}{\mu }_1$, ${\mu }_1={\text{cL}}_{{\text{la}}_2}{\mu }_2,>\mathrm{.}\mathrm{.}\mathrm{.}$, where ${\text{cL}}_{\text{la}}\mu =\mu \circ \text{ml}-\text{la}\mu$ and $\text{ml}:\text{T}\to \text{T}$ denotes the multiplication by $\mathrm{\ell }$ map. We show the following finiteness result: each Lipschitz function $\tau$ can be written in a canonical way as, $$\tau ={\text{cL}}_{{\text{la}}_1}\circ \mathrm{.}\mathrm{.}\mathrm{.}\circ {\text{cL}}_{{\text{la}}_m}\mu ,$$ where $m\ge 0$, $\lambda_1, …, \lambda_m \in (0, 1]$andtheLipschitzfunction$\mu$satisfies$\mu \neq \cL_\la \rho$foreverycontinuousfunction$\rho$andevery$\la \in (0,1]$.