Abstract
Letf be a two-dimensional area-preserving twist map. Given an irrational rotation number ω in the rotation interval off, there is an invariant recurrent set on whichf preserves the circular ordering and which has rotation number ω. For large nonlinearity, the parameter regime we are interested in, this set is a Cantor set. We show that well-ordered (minimizing) sets with rotation numbers close to ω are exponentially close to the Cantor set under study. The detailed configuration of well-ordered (minimizing) sets is universal and depends on one parameter, namely the Lyapunov coefficient of the Cantor set. There is a quantitative correspondence between this and similar behavior in the noninvertible circle map.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. I. Arnold,Geometrical Methods in the Theory of ODE's (Springer, 1986).
S. Aubry, inStructures et Instabilités, C. Godreche, ed. (Les Editions de Physique, 1986).
Q. Chen, J. D. Meiss, and I. C. Percival,Physica 29D:143–154 (1987).
D. Goroff,Erg. Theory Dyn. Syst. 5:337–339 (1985).
A. Katok,Erg. Theory Dyn. Syst. 2:185–194 (1982).
A. Katok, More about Birkhoff periodic and Mather sets for twistmaps, Maryland preprint (1982).
O. Lanford, inRegular and Chaotic Motions in Dynamic Systems, G. Velo and A. S. Wightman, eds. (Plenum Press, New York, 1985).
P. Le Calvez, Les ensembles de L'Aubry-Mather d'un difféomorphisme conservatif de l'anneau déviant la verticale sont en général hyperboliques, preprint (1987).
W. Li and M. Bak,Phys. Rev. Lett. 57(6):655–658 (1986).
R. S. MacKay, Introduction to dynamics of area preserving maps, Warwick preprint (1986).
R. S. MacKay, Exact results for an approximate renormalization⋯, Warwick preprint (1988).
R. S. MacKay, J. D. Meiss, and I. C. Percival,Phys. Rev. Lett. 52(9):697–700 (1984).
J. N. Mather,Publ. Math. IHES 63:153–204 (1986).
J. N. Mather, Destruction of invariant circles, preprint ETH Zurich (1987).
S. J. Shenker,Physica 5D:405–411 (1982).
F. M. Tangerman and J. J. P. Veerman, Asymptotic geometry of hyperbolic well-ordered Cantor sets, preprint (1989).
J. J. P. Veerman,Physica 134A:177–192 (1986).
J. J. P. Veerman,Physica 29D:191–201 (1987).
J. J. P. Veerman, Irrational rotation numbers, preprint (1987).
J. J. P. Veerman, Hausdorff dimension of order preserving sets, preprint (1988).
J. J. P. Veerman and F. M. Tangerman, On Aubry-Mather sets, preprint (1988a).
J. J. P. Veerman and F. M. Tangerman, Intersection properties of invariant manifolds in certain twist maps, preprint (1988b).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Veerman, J.J.P., Tangerman, F.M. Renormalization of Aubry-Mather Cantor sets. J Stat Phys 56, 83–98 (1989). https://doi.org/10.1007/BF01044233
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01044233