On the topological dynamics and phase-locking renormalization of Lorenz-like maps

Lluis Alsedà[1]; Antonio Falcó[2]

  • [1] Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028 Barcelona (Espagne)
  • [2] Universidad Cardenal Herrera-CEU, Facultad de Ciencias Sociales Jurídicas, Departamento de Economía y Empresa, Campus de Elche, Comissari 1, 03203 Elche-Elx (Espagne)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 3, page 859-883
  • ISSN: 0373-0956

Abstract

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The aim of this paper is twofold. First we give a characterization of the set of kneading invariants for the class of Lorenz–like maps considered as a map of the circle of degree one with one discontinuity. In a second step we will consider the subclass of the Lorenz– like maps generated by the class of Lorenz maps in the interval. For this class of maps we give a characterization of the set of renormalizable maps with rotation interval degenerate to a rational number, that is, of phase–locking renormalizable maps. This characterization is given by showing the equivalence between the geometric renormalization procedure and the combinatorial one (which is expressed in terms of an * –like product defined in the set of kneading invariants). Finally, we will prove the existence, at a combinatorial level, of periodic points of all periods for the renormalization map.

How to cite

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Alsedà, Lluis, and Falcó, Antonio. "On the topological dynamics and phase-locking renormalization of Lorenz-like maps." Annales de l’institut Fourier 53.3 (2003): 859-883. <http://eudml.org/doc/116056>.

@article{Alsedà2003,
abstract = {The aim of this paper is twofold. First we give a characterization of the set of kneading invariants for the class of Lorenz–like maps considered as a map of the circle of degree one with one discontinuity. In a second step we will consider the subclass of the Lorenz– like maps generated by the class of Lorenz maps in the interval. For this class of maps we give a characterization of the set of renormalizable maps with rotation interval degenerate to a rational number, that is, of phase–locking renormalizable maps. This characterization is given by showing the equivalence between the geometric renormalization procedure and the combinatorial one (which is expressed in terms of an $*$–like product defined in the set of kneading invariants). Finally, we will prove the existence, at a combinatorial level, of periodic points of all periods for the renormalization map.},
affiliation = {Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028 Barcelona (Espagne); Universidad Cardenal Herrera-CEU, Facultad de Ciencias Sociales Jurídicas, Departamento de Economía y Empresa, Campus de Elche, Comissari 1, 03203 Elche-Elx (Espagne)},
author = {Alsedà, Lluis, Falcó, Antonio},
journal = {Annales de l’institut Fourier},
keywords = {Lorenz maps; circle maps; kneading theory; renormalizable maps; periodic points},
language = {eng},
number = {3},
pages = {859-883},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the topological dynamics and phase-locking renormalization of Lorenz-like maps},
url = {http://eudml.org/doc/116056},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Alsedà, Lluis
AU - Falcó, Antonio
TI - On the topological dynamics and phase-locking renormalization of Lorenz-like maps
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 3
SP - 859
EP - 883
AB - The aim of this paper is twofold. First we give a characterization of the set of kneading invariants for the class of Lorenz–like maps considered as a map of the circle of degree one with one discontinuity. In a second step we will consider the subclass of the Lorenz– like maps generated by the class of Lorenz maps in the interval. For this class of maps we give a characterization of the set of renormalizable maps with rotation interval degenerate to a rational number, that is, of phase–locking renormalizable maps. This characterization is given by showing the equivalence between the geometric renormalization procedure and the combinatorial one (which is expressed in terms of an $*$–like product defined in the set of kneading invariants). Finally, we will prove the existence, at a combinatorial level, of periodic points of all periods for the renormalization map.
LA - eng
KW - Lorenz maps; circle maps; kneading theory; renormalizable maps; periodic points
UR - http://eudml.org/doc/116056
ER -

References

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