Please use this identifier to cite or link to this item: http://hdl.handle.net/2445/160863
 Title: Classification of linear skew-products of the complex plane and an affine route to fractalization Author: Fagella Rabionet, NúriaJorba i Monte, ÀngelJorba-Cuscó, MarcTatjer i Montaña, Joan Carles Keywords: Sistemes dinàmics diferenciablesFuncions de variables complexesDifferentiable dynamical systemsFunctions of complex variables Issue Date: Jul-2019 Publisher: American Institute of Mathematical Sciences (AIMS) Abstract: Linear skew products of the complex plane, \left.\begin{array}{l} \theta \mapsto \theta+\omega \\ z \mapsto a(\theta) z \end{array}\right\} where $\theta \in \mathrm{T}, z \in \mathbb{C}, \frac{\omega}{2 \pi}$ is irrational, and $\theta \mapsto a(\theta) \in \mathbb{C} \backslash\{0\}$ 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. Note: Versió postprint del document publicat a: https://doi.org/10.3934/dcds.2019153 It is part of: Discrete and Continuous Dynamical Systems-Series A, 2019, vol. 39, num. 7, p. 3767-3787 URI: http://hdl.handle.net/2445/160863 Related resource: https://doi.org/10.3934/dcds.2019153 ISSN: 1078-0947 Appears in Collections: Articles publicats en revistes (Matemàtiques i Informàtica)

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