On the Topology of Solenoidal Attractors of the Cylinder

Details On the Topology of Solenoidal Attractors of the Cylinder On the Topology of Solenoidal Attractors of the Cylinder

2004-03-30

arxiv.org/abs/math/0403523v1

Mathematics

Dynamical Systems

30 pages

Scientific paper

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 $\la \in (0,1]$.

Affiliated with

Also associated with

No associations

Rating

On the Topology of Solenoidal Attractors of the Cylinder does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the Topology of Solenoidal Attractors of the Cylinder, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Topology of Solenoidal Attractors of the Cylinder will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-78845