Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-13T01:48:59.954Z Has data issue: false hasContentIssue false
  • English
  • Français

Arithmetic Density

Part of: Diophantine approximation, transcendental number theory Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  14 December 2015

Mauricio Garay
Mauricio Garay*
Affiliation:
Lycée Franco-Allemand, Rue Colin Mamet, 78530 Buc, France (garay91@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We give measure estimates for sets appearing in the study of dynamical systems, such as preimages of Diophantine classes.

Keywords

Diophantine approximationKAM theoryHerman invariant tori conjecture

MSC classification

Secondary: 11J83: Metric theory 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 691 - 700
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Arnold, V. I., Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Usp. Mat. Nauk 18(5) (1963), 13–40.Google Scholar
2. Bruno, A. D., Analytic form of differential equations, I, Trans. Mosc. Math. Soc. 25 (1971), 131–288.Google Scholar
3. Garay, M. D., The Herman conjecture, Preprint (arxiv.org/abs/1206.1245; 2012).Google Scholar
4. Garay, M. D., The Herman conjecture, in Singularities, Oberwolfach Reports, Report 46, pp. 43–45 (European Mathematical Society, 2012).Google Scholar
5. Garay, M. D., An abstract KAM theorem, Mosc. Math. J. 14(4) (2013), 745–772.Google Scholar
6. Garay, M. D., Degenerations of invariant Lagrangian manifolds, J. Singularities 8 (2014), 50–67.Google Scholar
7. Herman, M. R., Some open problems in dynamical systems, in Proc. International Congress of Mathematicians, Berlin, Germany, 18–27 August 1998, Volume II, pp. 797–808 (Universität Bielefeld, Fakultät für Mathematick, 1998).Google Scholar
8. Kleinbock, D., Extremal subspaces and their submanifolds, Geom. Funct. Analysis 13(2) (2003), 437–466.CrossRefGoogle Scholar
9. Kleinbock, D. Y. and Margulis, G. A., Flows on homogeneous spaces and Diophantine approximation on manifolds, Annals Math. 148 (1998), 339–360.Google Scholar
10. Kolmogorov, A. N., On the conservation of quasi-periodic motions for a small perturbation of the Hamiltonian function, Dokl. Akad. Nauk SSSR 98 (1954), 527–530 (in Russian).Google Scholar
11. Mahler, K., Über das Maß der Menge aller S-Zahlen, Math. Annalen 106(1) (1932), 131–139.CrossRefGoogle Scholar
12. Pyartli, A. S., Diophantine approximations on submanifolds of Euclidean space, Funct. Analysis Applic. 3(4) (1969), 303–306.CrossRefGoogle Scholar
13. Rüssmann, H., Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Reg. Chaot. Dyn. 6(2) (2001), 119–204.Google Scholar
14. Sprindzhuk, V. G., A proof of Mahler's conjecture on the measure of the set of S-numbers, Izv. RAN Ser. Mat. 29(2) (1965), 379–436.Google Scholar