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Rotation orbits and the Farey tree

Published online by Cambridge University Press:  14 October 2010

Lisa Goldberg  and
Charles Tresser
Lisa Goldberg
Affiliation:
Barra Inc., 1995University Av., Berkeley CA94707USA
Charles Tresser
Affiliation:
I.B.M.Po Box 218, Yorktown Heights, NY 10598, USA
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Abstract

An α-rotation orbit for a self-map of the circle is an orbit whose cyclic order on the circle is the same as for any orbit of rotation by a. In this paper, we classify sets which are minimal sets comprised of a-rotation orbits for degree d αself-coverings of the circle.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 5 , October 1996 , pp. 1011 - 1029
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[Br]Brocot, A.. Calcul des Rouages par Approximations, Nouvelles Mélhodes. Paris, 1862.Google Scholar
[Cah]Cahen, E.. Eléments de la Théorie des Nombres. Gauthiers-Villars, Paris, 1900, pp. 331–335.Google Scholar
[Cau]Cauchy, A. L.. Démonstration d'un théorème curieux sur les nombres, Bull. Soc. Philomatique 1816. Oeuvres, Set. 2, Vol. 6. Gauthiers-Villars, Paris, 1882–1958, pp 146–148.Google Scholar
[CGT]Chenciner, A., Gambaudo, J. M. and Tresser, C.. Une remarque sur la structure des endomorphismes de degré I du cercle. C. R. Acad. Sci, Paris Série I t. 299 (1984), 145–148.Google Scholar
[Cv]Cvitanovié, P.. Farey organization of the fractal Hall effect. Physica Scripta T9 (1985), 202.CrossRefGoogle Scholar
[CSS]Cvitanovié, P., Shraiman, B. and Söderberg, B.. Scaling laws for mode locking in circle maps. Physica Scripta 32 (1985), 263–270.CrossRefGoogle Scholar
[Di]Dickson, L. E.. History of the Theory of Numbers, Vol. I. Chelsea, New York, 1952. (Or Carnegie Institution, 1919.)Google Scholar
[Fa1]Farey, J.. On a curious property of vulgar fractions. Phil. Mag. J. London 47 (1816), 385–386.Google Scholar
[Fa2]Farey, J.. Propriété curieuse des fractions ordinaires. Bull. Sci. Soc. Philomatique (3) 3 (1816), 112. (Page numbers 105–112 occur twice in the volume: this is the first page 112.)Google Scholar
[GGT]Gambaudo, J. M., Glendinning, P. and Tresser, C.. Collage de cycles et suites de Farey. C. R. Acad. Set, Paris Série i t. 299 (1984), 711–714.Google Scholar
[GT]Gambaudo, J. M. and Tresser, C.. On the dynamics of quasi-contractions. Boll. Soc. Braz. Math. 19 (1988), 61–114.CrossRefGoogle Scholar
[Go]Goldberg, L. R.. Fixed points of polynomial maps, part 1: rotation subsets. Ann. Sci. Éc. Norm. Sup. 25 (1992), 679–685.CrossRefGoogle Scholar
[GM]Goldberg, L. R. and Milnor, J.. Fixed points of polynomial maps, part II: fixed points portraits. Ann. Sci. Éc. Norm. Sup. 26 (1993), 51–98.CrossRefGoogle Scholar
[HW]Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers. Fifth edn.Clarendon, Oxford, 1979.Google Scholar
[Har]Haros, Le C.en, TABLES por évaluer une fraction ordinaire avec autan de décimales qu'on voudra; et pour trouver la fraction ordinaire la plus simples, et qui approche sensiblement d'une fraction décimale. Annonces et Notices d'Ouvrages. Jour, de l'Ecole Polytechnique Cah. 11, t. 4 (1802), 364–368.Google Scholar
[He]Hedlund, G. A.. Sturmian minimal sets. Amer. J. Math. 66 (1944), 605–620.CrossRefGoogle Scholar
[Hu]Humbert, G.. Sur les fractions continues ordinaires et les formes quadratiques binaires indefinies. J. Math. Pures Apll 2 (1916), 104–154.Google Scholar
[La]Lagarias, J. C.. Number theory and dynamical systems. Proc. Symp. Appl. Math. 46 (1992), 35–72.CrossRefGoogle Scholar
[LT]Lagarias, J. C. and Tresser, C.. A walk along the branches of the extended Farey tree. IBM Journal of R&D. 39 (1995), 283–294.CrossRefGoogle Scholar
[Lu]Lucas, E., Théorie des Nombres, Vol. 1. Gauthiers-Villars, Paris, 1891, pp. 467–475 and 508–510.Google Scholar
[MH]Morse, M. and Hedlund, G. A.. Symbolic dynamics II: Sturmian trajectories. Amer. J. Math. 62 (1940), 1–42.CrossRefGoogle Scholar
[OK]Ostlund, S. and Kim, S. H.. Renormalization of quasiperiodic mappings. Physica Scripta T9 (1985), 193–198.CrossRefGoogle Scholar
[Po]Poincaré, H.. Sur les courbes définies par des équations différentielles. J. Math. Pures Appl. Aeme série, 1 (1885), 167–244. Also in Œuvres Complètes, t. 1. Gauthier-Villars, Paris, 1951.Google Scholar
[STZ]Siegel, R., Tresser, C. and Zettler, G.. A decoding problem in dynamics and in number theory. Chaos 2 (1992), 473–193.CrossRefGoogle ScholarPubMed
[Sin]Smith, H. J. S.. Note on continued fractions. The Messenger of Mathematics VI (1877), 51–14.Google Scholar