A fractalization process for affine skew-products on the complex plane

CDSNS Colloquium
Wednesday, May 10, 2017 - 13:00
1 hour (actually 50 minutes)
Skiles 005
Universitat de Barcelona
Consider an affine skew product of the complex plane. \begin{equation}\begin{cases}        \omega \mapsto \theta+\omega,\\        z \mapsto =a(\theta  \mu)z+c, \end{cases}\end{equation}where $\theta \in \mathbb{T}$, $z\in \mathbb{C}$, $\omega$ is Diophantine, and $\mu$ and $c$ are real parameters. In this talk we show that, under suitable conditions, the affine skew product has an invariant curve that undergoes a fractalization process when $\mu$ goes to a critical value. The main hypothesis needed is the lack of reducibility of the system.  A characterization of reducibility of linear skew-products on the complex plane is provided. We also include a linear and topological classification of these systems. Join work with: N\'uria Fagella, \`Angel Jorba and Joan Carles Tatjer