Classification of linear skew-products of the complex plane and an affine route to fractalization

Abstract

Linear skew products of the complex plane, \begin{document}$ \left. \begin{array}{rcl} \theta & \mapsto & \theta+\omega,\\ z & \mapsto & a(\theta)z, \end{array} \right\} $\end{document} where $ \theta\in {\mathbb T} $, $ z\in {\mathbb C} $, $ \frac{\omega}{2\pi} $ is irrational, and $ \theta\mapsto a(\theta) \in {\mathbb C}\setminus \{0\} $ is a smooth map, appear naturally when linearizing dynamics around an invariant curve of a quasi-periodically forced complex map. In this paper we study linear and topological equivalence classes of such maps through conjugacies which preserve the skewed structure, relating them to the Lyapunov exponent and the winding number of $ \theta\mapsto a(\theta) $. We analyze the transition between these classes by considering one parameter families of linear skew products. Finally, we show that, under suitable conditions, an affine variation of the maps above has a non-reducible invariant curve that undergoes a fractalization process when the parameter goes to a critical value. This phenomenon of fractalization of invariant curves is known to happen in nonlinear skew products, but it is remarkable that it also occurs in simple systems as the ones we present.