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On the measure of KAM tori in two degrees of freedom
Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo San L. Murialdo 1 - 00146 Roma, Italy |
A conjecture of Arnold, Kozlov and Neishtadt on the exponentially small measure of the "non-torus" set in analytic systems with two degrees of freedom is discussed.
References:
| [1] |
V. I. Arnold,
Conditions for the applicability, and estimate of the error, of an averaging method for systems which pass through states of resonance in the course of their evolution, Collected Works, 1 (1965), 477-480.
doi: 10.1007/978-3-642-01742-1_31. |
| [2] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 2006. |
| [3] |
L. Biasco and L. Chierchia,
On the measure of Lagrangian invariant tori in nearly–integrable mechanical systems, Rend. Lincei Mat. Appl., 26 (2015), 423-432.
doi: 10.4171/RLM/713. |
| [4] |
L. Biasco and L. Chierchia, KAM Theory for secondary tori, arXiv: 1702.06480v1 [math.DS]. |
| [5] |
L. Biasco and L. Chierchia,
Explicit estimates on the measure of primary KAM tori, Ann. Mat. Pura Appl. (4), 197 (2018), 261-281.
doi: 10.1007/s10231-017-0678-8. |
| [6] |
L. Biasco and L. Chierchia, On the topology of nearly–integrable Hamiltonians at simple resonances., To appear in Nonlinearity, 2020. arXiv: 1907.09434 [math.DS] |
| [7] |
L. Biasco and L. Chierchia, Exponentially small measure of the non–torus set in 2 degrees of freedom, Work in progress. |
| [8] |
P. A. M. Dirac,
The adiabatic invariance of the quantum integrals, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 107 (1925), 725-734.
doi: 10.1098/rspa.1925.0052. |
| [9] |
M. Guzzo, L. Chierchia and G. Benettin,
The steep Nekhoroshev Theorem, Commun. Math. Phys., 342 (2016), 569-601.
doi: 10.1007/s00220-015-2555-x. |
| [10] |
B.R. Hunt, T. Sauer and J.A. Yorke,
Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 217-238.
doi: 10.1090/S0273-0979-1992-00328-2. |
| [11] |
B. R. Hunt and V. Y. Kaloshin, Prevalence, chapter 2, Handbook in Dynamical Systems, edited by H. Broer, F. Takens, B. Hasselblatt, 3 (2010), 43–87. |
| [12] |
A. G. Medvedev, A. I. Neishtadt and D. V. Treschev,
Lagrangian tori near resonances of near–integrable Hamiltonian systems, Nonlinearity, 28 (2015), 2105-2130.
doi: 10.1088/0951-7715/28/7/2105. |
| [13] |
A. I. Neishtadt,
On passage through resonances in the two-frequency problem, Sov. Phys., Dokl., 20 (1975), 189-191.
|
| [14] |
A. I. Neishtadt,
Averaging, passage through resonances, and capture into resonance in two–frequency systems, Russian Math. Surveys, 69 (2014), 771-843.
doi: 10.4213/rm9603. |
| [15] |
N. N. Nekhoroshev,
An exponential estimate of the time of stability of nearly- integrable Hamiltonian systems I, Math. Surveys, 32 (1977), 1-65.
|
| [16] |
J. Pöschel,
Nekhoroshev estimates for quasi–convex Hamiltonian systems, Math. Z., 213 (1993), 187-216.
doi: 10.1007/BF03025718. |
show all references
References:
| [1] |
V. I. Arnold,
Conditions for the applicability, and estimate of the error, of an averaging method for systems which pass through states of resonance in the course of their evolution, Collected Works, 1 (1965), 477-480.
doi: 10.1007/978-3-642-01742-1_31. |
| [2] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 2006. |
| [3] |
L. Biasco and L. Chierchia,
On the measure of Lagrangian invariant tori in nearly–integrable mechanical systems, Rend. Lincei Mat. Appl., 26 (2015), 423-432.
doi: 10.4171/RLM/713. |
| [4] |
L. Biasco and L. Chierchia, KAM Theory for secondary tori, arXiv: 1702.06480v1 [math.DS]. |
| [5] |
L. Biasco and L. Chierchia,
Explicit estimates on the measure of primary KAM tori, Ann. Mat. Pura Appl. (4), 197 (2018), 261-281.
doi: 10.1007/s10231-017-0678-8. |
| [6] |
L. Biasco and L. Chierchia, On the topology of nearly–integrable Hamiltonians at simple resonances., To appear in Nonlinearity, 2020. arXiv: 1907.09434 [math.DS] |
| [7] |
L. Biasco and L. Chierchia, Exponentially small measure of the non–torus set in 2 degrees of freedom, Work in progress. |
| [8] |
P. A. M. Dirac,
The adiabatic invariance of the quantum integrals, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 107 (1925), 725-734.
doi: 10.1098/rspa.1925.0052. |
| [9] |
M. Guzzo, L. Chierchia and G. Benettin,
The steep Nekhoroshev Theorem, Commun. Math. Phys., 342 (2016), 569-601.
doi: 10.1007/s00220-015-2555-x. |
| [10] |
B.R. Hunt, T. Sauer and J.A. Yorke,
Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 217-238.
doi: 10.1090/S0273-0979-1992-00328-2. |
| [11] |
B. R. Hunt and V. Y. Kaloshin, Prevalence, chapter 2, Handbook in Dynamical Systems, edited by H. Broer, F. Takens, B. Hasselblatt, 3 (2010), 43–87. |
| [12] |
A. G. Medvedev, A. I. Neishtadt and D. V. Treschev,
Lagrangian tori near resonances of near–integrable Hamiltonian systems, Nonlinearity, 28 (2015), 2105-2130.
doi: 10.1088/0951-7715/28/7/2105. |
| [13] |
A. I. Neishtadt,
On passage through resonances in the two-frequency problem, Sov. Phys., Dokl., 20 (1975), 189-191.
|
| [14] |
A. I. Neishtadt,
Averaging, passage through resonances, and capture into resonance in two–frequency systems, Russian Math. Surveys, 69 (2014), 771-843.
doi: 10.4213/rm9603. |
| [15] |
N. N. Nekhoroshev,
An exponential estimate of the time of stability of nearly- integrable Hamiltonian systems I, Math. Surveys, 32 (1977), 1-65.
|
| [16] |
J. Pöschel,
Nekhoroshev estimates for quasi–convex Hamiltonian systems, Math. Z., 213 (1993), 187-216.
doi: 10.1007/BF03025718. |
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