On the topological dynamics and phase-locking renormalization of Lorenz-like maps
[Dynamique topologique et renormalisation des applications de type Lorenz]
Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 859-883.

L’objectif de cet article est double. D’abord, on donne une caractérisation de l’ensemble des invariants de pétrissage pour la classe des applications de type Lorenz considérées comme des applications du cercle de degré un avec une discontinuité. Dans une deuxième étape, on considère la sous-classe des applications de type Lorenz engendrées pour la classe des applications de Lorenz dans l’intervalle. Pour cette classe des applications on donne une caractérisation de l’ensemble des applications renormalisables avec intervalle de rotation dégénéré à un numéro rationnel, c’est-à-dire des applications de confinement de phase renormalisables. On obtient cette caractérisation en montrant l’équivalence entre la méthode de renormalisation géométrique et la méthode combinatoire (qu’on exprime en termes d’un produit du type * défini sur l’ensemble des invariants de pétrissage. Finalement, on démontre l’existence au niveau combinatoire, de points périodiques de toutes les périodes de l’application de renormalisation.

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.

DOI : 10.5802/aif.1963
Classification : 37E10, 37E20
Keywords: Lorenz maps, circle maps, kneading theory
Mot clés : applications de Lorenz, applications du cercle, théorie du pétrissage
Alsedà, Lluis 1 ; Falcó, Antonio 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)
@article{AIF_2003__53_3_859_0,
     author = {Alsed\`a, Lluis and Falc\'o, Antonio},
     title = {On the topological dynamics and phase-locking renormalization of {Lorenz-like} maps},
     journal = {Annales de l'Institut Fourier},
     pages = {859--883},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {3},
     year = {2003},
     doi = {10.5802/aif.1963},
     mrnumber = {2008444},
     zbl = {1027.37021},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1963/}
}
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
SP  - 859
EP  - 883
VL  - 53
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1963/
DO  - 10.5802/aif.1963
LA  - en
ID  - AIF_2003__53_3_859_0
ER  - 
%0 Journal Article
%A Alsedà, Lluis
%A Falcó, Antonio
%T On the topological dynamics and phase-locking renormalization of Lorenz-like maps
%J Annales de l'Institut Fourier
%D 2003
%P 859-883
%V 53
%N 3
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1963/
%R 10.5802/aif.1963
%G en
%F AIF_2003__53_3_859_0
Alsedà, Lluis; Falcó, Antonio. On the topological dynamics and phase-locking renormalization of Lorenz-like maps. Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 859-883. doi : 10.5802/aif.1963. http://www.numdam.org/articles/10.5802/aif.1963/

[1] Ll. Alsedà; A. Falcó A characterization of the kneading pair for bimodal degree one circle maps, Ann. Inst. Fourier, Volume 47 (1997) no. 1, pp. 273-304 | DOI | Numdam | MR | Zbl

[2] Ll. Alsedà; J. Llibre; M. Misiurewicz Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, 5, World Scientific, Singapore, 1993 | MR | Zbl

[3] Ll. Alsedà; J. Llibre; M. Misiurewicz; Ch. Tresser Periods and entropy for Lorentz-like maps, Ann. Inst. Fourier, Volume 39 (1989) no. 4, pp. 929-952 | DOI | Numdam | MR | Zbl

[4] Ll. Alsedà; F. Mañosas Kneading theory and rotation interval for a class of circle maps of degree one, Nonlinearity, Volume 3 (1990), pp. 413-452 | DOI | MR | Zbl

[5] Ll. Alsedà; F. Mañosas Kneading theory for a family of circle maps with one discontinuity, Acta Math. Univ. Comeniae, Volume LXV (1996), pp. 11-22 | MR | Zbl

[6] P. Collet; J.P. Eckmann Iterated maps on the interval as dynamical systems, Progress in Physics, Birkhäuser, 1980 | MR | Zbl

[7] A. Falcó Bifurcations and symbolic dynamics for bimodal degree one circle maps: The Arnol'd tongues and the Devil's staircase (1995) (Ph.D. Thesis, Universitat Autònoma de Barcelona)

[8] P. Glendinning Topological conjugation of Lorenz maps by β-transformations, Math. Proc. Camb. Phil. Soc., Volume 107 (1990), pp. 401-413 | DOI | MR | Zbl

[9] P. Glendinning; C. Sparrow Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps, Physica D, Volume 62 (1993), pp. 22-50 | DOI | MR | Zbl

[10] J. Gukenheimer A strange, strange attractor. The Hopf bifurcations and its applications, Appl. Math. Sci., 19, Springer-Verlag, 1976

[11] J. Gukenheimer; R.F. Williams Structural stability of Lorenz attractors, Publ. Math. IHES, Volume 50 (1979), pp. 307-320 | Numdam | Zbl

[12] J.H. Hubbard; C. Sparrow The Classification of Topologically Expansive Lorenz Maps, Comm. Pure Appl. Math., Volume XLIII (1990), pp. 431-443 | DOI | MR | Zbl

[13] E.N. Lorenz Deterministic non-periodic flow, J. Atmos. Sci., Volume 20 (1963), pp. 130-141 | DOI

[14] M. Martens The periodic points of renormalization, Ann. of Math., Volume 147 (1998), pp. 543-584 | DOI | MR | Zbl

[15] M. Martens; W. de Melo Universal models for Lorenz maps (1997) (Preprint, IMPA)

[16] J. Milnor; P. Thurston On iterated maps on the interval I, II, Dynamical Systems (Lecture Notes in Math.), Volume 1342 (1988) | Zbl

[17] M. Misiurewicz Rotation intervals for a class of maps of the real line into itself, Ergod. Th. \& Dynam. Sys., Volume 6 (1986), pp. 117-132 | MR | Zbl

[18] C.A. Morales; E.R. Pujals Singular strange attractors on the boundary of Morse-Smale systems, Ann. Sci. École Norm. Sup., Volume 30 (1997), pp. 693-717 | Numdam | MR | Zbl

[19] H. Poincaré Sur les courbes définies par les équations différentielles, Œuvres complètes, Volume vol. 1 (1952), pp. 137-158

[20] F. Rhodes; C.L. Thompson Rotation numbers for monotone functions on the circle, J. London Math. Soc., Volume 34 (1986), pp. 360-368 | DOI | MR | Zbl

[21] C.T. Sparrow The Lorenz equations: Bifurcations, chaos and strange attractors, Appl. Math. Sci., 41, Springer-Verlag, 1982 | MR | Zbl

[22] R.F. Williams The structure of Lorenz attractors, Publ. Math. IHES, Volume 50 (1979), pp. 321-347 | Numdam | MR | Zbl

Cité par Sources :