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

Núria Fagella; Àngel Jorba; Marc Jorba-Cuscó; Joan Carles Tatjer

doi:10.3934/dcds.2019153

Article Contents

2019, Volume 39, Issue 7: 3767-3787. Doi: 10.3934/dcds.2019153

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

Departament de Matemàtiques i Informàtica, Barcelona Graduate School of Mathematics (BGSMath), Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain

Received: February 2018

Revised: November 2018

Published: April 2019

Work supported by the Maria de Maeztu Excellence Grant MDM-2014-0445 and the grant 2017 SGR 1374. N. Fagella has been partially supported by the grants MTM2014-52209-C2-2-P and MTM2017-86795-C3-3-P, A. Jorba, M. Jorba-Cuscó and J.C. Tatjer have been supported by the grant MTM2015-67724-P.

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Abstract

Linear skew products of the complex plane,

$ \left. \begin{array}{rcl} \theta & \mapsto & \theta+\omega,\\ z & \mapsto & a(\theta)z, \end{array} \right\} $

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.

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Mathematics Subject Classification: Primary: 37C60; Secondary: 30D05, 37D25.

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Figure 1. Invariant curve of (3) for c = 1. Plots for µ = 0:5, µ = 0:9, µ = 0:99 and µ = 0:999

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Figure 2. Asymptotic growth of the invariant curve of (3) w.r.t. $\mu$ when $\mu\nearrow 1$, for $c = 1$. The horizontal axis shows $1-\mu$ and the symbols "+'' denote the computed values. The dotted line is the fitting function. Top: On the left, fitting $\|z_{\mu}\|_{\infty}$ by $1.54(1-\mu)^{-1/2}$. On the right, fitting of $\|z_{\mu}'\|_{\infty}$ by $0.41(1-\mu)^{-3/2}$. Bottom: On the left, fitting of the length of $z_{\mu}$ by $3.1(1-\mu)^{-3/2}$. On the right, fitting of $(z_{\mu}, 0)$ by $0.5(1-\mu)^{-1}$

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