Classification of linear skew-products of the complex plane and an affine route to fractalization
Fagella Rabionet, Núria (Universitat de Barcelona. Departament de Matemàtiques i Informàtica)
Jorba, Àngel (Universitat de Barcelona. Departament de Matemàtiques i Informàtica)
Jorba-Cuscó, Marc (Universitat de Barcelona. Departament de Matemàtiques i Informàtica)
Tatjer, Joan Carles (Universitat de Barcelona. Departament de Matemàtiques i Informàtica)

Date:	2019
Abstract:	Linear skew products of the complex plane, θ↦θ+ω,z↦a(θ)z,} where θ∈T, z∈C, ω/2π is irrational, and [θ↦a(θ)∈C∖{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 θ↦a(θ). 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.
Grants:	Ministerio de Economía y Competitividad MDM-2014-0445 Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1374 Ministerio de Economía y Competitividad MTM2015-67724-P Ministerio de Economía y Competitividad MTM2014-52209-C2-2-P Ministerio de Economía y Competitividad MTM2017-86795-C3-3-P
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Reducibility ; Winding number ; Lyapunov exponent ; Complex fibered maps ; Topological classification
Published in:	Discrete and continuous dynamical systems. Series A, Vol. 39, Issue 7 (July 2019) , p. 3767-3787, ISSN 1553-5231

DOI: 10.3934/dcds.2019153

Postprint 22 p, 1.6 MB

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