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

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 m...

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Autores: Fagella Rabionet, Núria, Jorba i Monte, Àngel, Jorba-Cuscó, Marc, Tatjer i Montaña, Joan Carles
Tipo de recurso: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2019
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2445/160863
Acceso en línea:https://hdl.handle.net/2445/160863
Access Level:acceso abierto
Palabra clave:Sistemes dinàmics diferenciables
Funcions de variables complexes
Differentiable dynamical systems
Functions of complex variables
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spelling Classification of linear skew-products of the complex plane and an affine route to fractalizationFagella Rabionet, NúriaJorba i Monte, ÀngelJorba-Cuscó, MarcTatjer i Montaña, Joan CarlesSistemes dinàmics diferenciablesFuncions de variables complexesDifferentiable dynamical systemsFunctions of complex variablesLinear 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.American Institute of Mathematical Sciences (AIMS)2020202020192020info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersion21 p.application/pdfapplication/pdfhttps://hdl.handle.net/2445/160863Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésVersió postprint del document publicat a: https://doi.org/10.3934/dcds.2019153Discrete and Continuous Dynamical Systems-Series A, 2019, vol. 39, num. 7, p. 3767-3787https://doi.org/10.3934/dcds.2019153(c) American Institute of Mathematical Sciences (AIMS), 2019info:eu-repo/semantics/openAccessoai:recercat.cat:2445/1608632026-05-29T05:05:01Z
dc.title.none.fl_str_mv Classification of linear skew-products of the complex plane and an affine route to fractalization
title Classification of linear skew-products of the complex plane and an affine route to fractalization
spellingShingle Classification of linear skew-products of the complex plane and an affine route to fractalization
Fagella Rabionet, Núria
Sistemes dinàmics diferenciables
Funcions de variables complexes
Differentiable dynamical systems
Functions of complex variables
title_short Classification of linear skew-products of the complex plane and an affine route to fractalization
title_full Classification of linear skew-products of the complex plane and an affine route to fractalization
title_fullStr Classification of linear skew-products of the complex plane and an affine route to fractalization
title_full_unstemmed Classification of linear skew-products of the complex plane and an affine route to fractalization
title_sort Classification of linear skew-products of the complex plane and an affine route to fractalization
dc.creator.none.fl_str_mv Fagella Rabionet, Núria
Jorba i Monte, Àngel
Jorba-Cuscó, Marc
Tatjer i Montaña, Joan Carles
author Fagella Rabionet, Núria
author_facet Fagella Rabionet, Núria
Jorba i Monte, Àngel
Jorba-Cuscó, Marc
Tatjer i Montaña, Joan Carles
author_role author
author2 Jorba i Monte, Àngel
Jorba-Cuscó, Marc
Tatjer i Montaña, Joan Carles
author2_role author
author
author
dc.subject.none.fl_str_mv Sistemes dinàmics diferenciables
Funcions de variables complexes
Differentiable dynamical systems
Functions of complex variables
topic Sistemes dinàmics diferenciables
Funcions de variables complexes
Differentiable dynamical systems
Functions of complex variables
description 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.
publishDate 2019
dc.date.none.fl_str_mv 2019
2020
2020
2020
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/160863
url https://hdl.handle.net/2445/160863
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Versió postprint del document publicat a: https://doi.org/10.3934/dcds.2019153
Discrete and Continuous Dynamical Systems-Series A, 2019, vol. 39, num. 7, p. 3767-3787
https://doi.org/10.3934/dcds.2019153
dc.rights.none.fl_str_mv (c) American Institute of Mathematical Sciences (AIMS), 2019
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) American Institute of Mathematical Sciences (AIMS), 2019
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 21 p.
application/pdf
application/pdf
dc.publisher.none.fl_str_mv American Institute of Mathematical Sciences (AIMS)
publisher.none.fl_str_mv American Institute of Mathematical Sciences (AIMS)
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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