Classification of linear skew-products of the complex plane and an affine route to fractalization (Q2423642)

The authors study linear skew-products on \(\mathbb{C}\) of the form \[ \theta \mapsto \theta+\omega, \quad z\mapsto a(\theta)z, \] where \(\theta \in \mathbb{T}\), \(\omega/2\pi\) is irrational, and \(a(\theta)\) is smooth. Also one-parameter versions of the form \[ \theta\mapsto \theta+\omega, \quad z\mapsto a(\theta,\mu)z, \] and affine variants of the above systems are considered. In the paper, these systems are classified using Lyapunov exponents and winding numbers, and the dependence of these quantities on the parameter \(\mu\) is investigated. A natural problem in this context is understanding how invariant curves of such systems behave close to a parameter value for which the systems admit no invariant curves. This question is addressed in some details, leading to a study of the so-called ``fractalization phenomena'', illustrated here both with images and rigorous results. At a first glance, the skew-products considered in this paper might look rather simple, but as the authors show, they in fact do exhibit a fair amount of complexity and interesting behavior.