A characterization of the kneading pair for bimodal degree one circle maps
Annales de l'Institut Fourier, Volume 47 (1997) no. 1, pp. 273-304.

For continuous maps on the interval with finitely many monotonicity intervals, the kneading theory developed by Milnor and Thurston gives a symbolic description of the dynamics of a given map. This description is given in terms of the kneading invariants which essentially consists in the symbolic orbits of the turning points of the map under consideration. Moreover, this theory also describes a classification of all such maps through theses invariants. For continuous bimodal degree one circle maps, similar invariants were introduced by Alsedà and Mañosas, where the first part of the program just described was carried through, and where relations between the circle maps invariants and the rotation interval were elucidated. The main theorem of this paper characterizes the set of kneading invariants for all bimodal degree one circle maps.

La théorie des itinéraires symboliques développée par Milnor et Thurston donne, pour les applications de l’intervalle dans lui-même avec un nombre fini de morceaux monotones, une caractérisation de la dynamique de ces applications. Dans cet article nous apportons une caractérisation de toutes les “paires de pétrissage” pour l’ensemble des relèvements des applications continues du cercle dans lui-même de degré un bimodales.

@article{AIF_1997__47_1_273_0,
     author = {Alsed\`a, Lluis and Falc\'o, Antonio},
     title = {A characterization of the kneading pair for bimodal degree one circle maps},
     journal = {Annales de l'Institut Fourier},
     pages = {273--304},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {1},
     year = {1997},
     doi = {10.5802/aif.1567},
     zbl = {0861.58014},
     mrnumber = {98h:58055},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1567/}
}
TY  - JOUR
AU  - Alsedà, Lluis
AU  - Falcó, Antonio
TI  - A characterization of the kneading pair for bimodal degree one circle maps
JO  - Annales de l'Institut Fourier
PY  - 1997
DA  - 1997///
SP  - 273
EP  - 304
VL  - 47
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1567/
UR  - https://zbmath.org/?q=an%3A0861.58014
UR  - https://www.ams.org/mathscinet-getitem?mr=98h:58055
UR  - https://doi.org/10.5802/aif.1567
DO  - 10.5802/aif.1567
LA  - en
ID  - AIF_1997__47_1_273_0
ER  - 
%0 Journal Article
%A Alsedà, Lluis
%A Falcó, Antonio
%T A characterization of the kneading pair for bimodal degree one circle maps
%J Annales de l'Institut Fourier
%D 1997
%P 273-304
%V 47
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1567
%R 10.5802/aif.1567
%G en
%F AIF_1997__47_1_273_0
Alsedà, Lluis; Falcó, Antonio. A characterization of the kneading pair for bimodal degree one circle maps. Annales de l'Institut Fourier, Volume 47 (1997) no. 1, pp. 273-304. doi : 10.5802/aif.1567. http://www.numdam.org/articles/10.5802/aif.1567/

[1] Ll. Alsedà, A. Falcó, An entropy formula for a class of circle maps, C. R. Acad. Sci. Paris, 314, Série I (1992), 677-682. | MR | Zbl

[2] Ll. Alsedà, A. Falcó, Devil's staircase route to chaos in a forced relaxation oscillator, Ann. Inst. Fourier, 44-1 (1994), 109-128. | Numdam | MR | Zbl

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

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

[5] Ll. Alsedà, F. Mañosas, W. Szlenk, A Characterization of the uniquely ergodic endomorphisms of the circle, Proc. Amer. Math. Soc., 117 (1993), 711-714. | MR | Zbl

[6] P.L. Boyland, Bifurcations of circle maps : Arnol'd Tongues, bistability and rotation intervals, Commun. Math. Phys., 106 (1986), 353-381. | MR | Zbl

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

[8] A. Chenciner, J.M. Gambaudo, Ch. Tresser, Une remarque sur la structure des endomorphismes de degré 1 du cercle, C. R. Acad. Sci., Paris series I, 299 (1984), 145-148. | MR | Zbl

[9] M. Denker, C. Grillenberger, K. Sigmund, Ergodic Theory in compact spaces, Lecture Notes in Math., 527, Springer, Berlin, 1976. | MR | Zbl

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

[11] L. Glass, M.C. Mackey, From clocks to chaos, Princeton University Press, 1988. | MR | Zbl

[12] R. Ito, Rotation sets are closed, Math. Proc. Camb. Phil. Soc., 89 (1981), 107-111. | MR | Zbl

[13] M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Mem. Amer. Math. Soc., 244 (1981). | MR | Zbl

[14] I. Malta, On Denjoy's theorem for endomorphisms, Ergod. Th. & Dynam. Sys., 6 (1986), 259-264. | MR | Zbl

[15] W. De Melo, S. Van Strien, One dimensional dynamics, Springer-Verlag, 1993. | MR | Zbl

[16] J. Milnor, P. Thurston, On iterated maps on the interval, I, II, Dynamical Systems, Lecture Notes in Math. 1342, Springer, (1988), 465-563. | Zbl

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

[18] S.E. Newhouse, J. Palis, F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci., Publ. Math., 57 (1983), 5-71. | Numdam | MR | Zbl

[19] H. Poincaré, Sur les curves définies par les équations differentielles, Œuvres complètes, vol 1, 137-158, Gauthiers-Villars, Paris, 1952.

[20] J.C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris, 298, Série I (1984), 141-144. | MR | Zbl

Cited by Sources: