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Gevrey normal form and effective stability of Lagrangian tori
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Convergence radius in the Poincaré-Siegel problem
Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''
1. | Department of Mathematics, Hunan Normal University, Changsha 410081, China |
2. | School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 30332-0160, United States |
We present examples showing that the answer cannot be based just on the size of the derivatives but that it also has to include considerations of the geometry of the domains.
We also present some sufficient conditions on the geometry of the domains that ensure that indeed the sum of the derivatives is a Whitney derivative of the sum of the functions.
References:
[1] |
R. Abraham and J. Robbin, "Transversal Mappings and Flows,'', W.A. Benjamin, (1967).
|
[2] |
V. I. Arnol$'$d, Proof of a theorem by A.N. Kolmogorov on the persistence of conditionally periodic motions under a small perturbation of the Hamiltonian,, Russian Math. Surveys, 18 (1963), 9.
doi: 10.1070/RM1963v018n05ABEH004130. |
[3] |
L. Chierchia and G. Gallavotti, Smooth prime integrals for quasi-integrable Hamiltonian systems,, Nuovo Cimento B (11), 67 (1982), 277.
doi: 10.1007/BF02721167. |
[4] |
R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems,, Comm. Math. Phys., 150 (1992), 289.
doi: 10.1007/BF02096662. |
[5] |
R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions,, Discrete Contin. Dynam. Systems, 5 (1999), 157.
|
[6] |
C. Fefferman, Interpolation and extrapolation of smooth functions by linear operators,, Rev. Mat. Iberoam., 21 (2005), 313.
|
[7] |
C. Fefferman, The structure of linear extension operators for $C^m$,, Rev. Mat. Iberoam., 23 (2007), 269.
|
[8] |
C. Fefferman, $C^m$ extension by linear operators,, Ann. of Math. (2), 166 (2007), 779.
doi: 10.4007/annals.2007.166.779. |
[9] |
C. Fefferman, Extension of $C^{m,\omega}$-smooth functions by linear operators,, Rev. Mat. Iberoam., 25 (2009), 1.
|
[10] |
C. Fefferman, Whitney's extension problems and interpolation of data,, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 207.
|
[11] |
G. Gallavotti, Perturbation theory for classical Hamiltonian systems,, in, 7 (1981), 359.
|
[12] |
L. Grafakos, "Classical and Modern Fourier Analysis,'', Pearson Education, (2004).
|
[13] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'', Encyclopedia of Mathematics and its Applications, 54 (1995).
|
[14] |
M. Nicol and A. Török, Whitney regularity for solutions to the coboundary equation on Cantor sets,, Math. Phys. Electron. J., 13 (2007).
|
[15] |
G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I. Birkhoff normal forms,, Ann. Henri Poincaré, 1 (2000), 223.
doi: 10.1007/PL00001004. |
[16] |
G. Popov, KAM theorem for Gevrey Hamiltonians,, Ergodic Theory Dynam. Systems, 24 (2004), 1753.
doi: 10.1017/S0143385704000458. |
[17] |
G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians,, Mat. Contemp., 26 (2004), 87.
|
[18] |
J. Pöschel, Integrability of Hamiltonian systems on Cantor sets,, Comm. Pure Appl. Math., 35 (1982), 653.
doi: 10.1002/cpa.3160350504. |
[19] |
J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems,, Math. Z., 202 (1989), 559.
doi: 10.1007/BF01221590. |
[20] |
M. Shub, "Global Stability of Dynamical Systems,'', Springer-Verlag, (1987).
|
[21] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'', Princeton Mathematical Series, 30 (1970).
|
[22] |
J. A. Vano, "A Nash-Moser Implicit Function Theorem with Whitney Regularity and Applications,'', Ph.D thesis, (2002), 02. Google Scholar |
[23] |
F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma,, Dyn. Syst., 18 (2003), 159.
doi: 10.1080/1468936031000117857. |
[24] |
H. Whitney, Analytic extensions of differentiable functions defined in closed sets,, Trans. Amer. Math. Soc., 36 (1934), 63.
|
[25] |
H. Whitney, Differentiable functions defined in arbitrary subsets of Euclidean space,, Trans. Amer. Math. Soc., 40 (1936), 309.
|
[26] |
J. Xu and J. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition,, J. Differential Equations, 235 (2007), 609.
doi: 10.1016/j.jde.2006.12.001. |
[27] |
X. Yuan, Construction of quasi-periodic breathers via KAM technique,, Comm. Math. Phys., 226 (2002), 61.
doi: 10.1007/s002200100593. |
[28] |
X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations,, J. Differential Equations, 230 (2006), 213.
doi: 10.1016/j.jde.2005.12.012. |
[29] |
E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I,, Comm. Pure Appl. Math., 28 (1975), 91.
|
show all references
References:
[1] |
R. Abraham and J. Robbin, "Transversal Mappings and Flows,'', W.A. Benjamin, (1967).
|
[2] |
V. I. Arnol$'$d, Proof of a theorem by A.N. Kolmogorov on the persistence of conditionally periodic motions under a small perturbation of the Hamiltonian,, Russian Math. Surveys, 18 (1963), 9.
doi: 10.1070/RM1963v018n05ABEH004130. |
[3] |
L. Chierchia and G. Gallavotti, Smooth prime integrals for quasi-integrable Hamiltonian systems,, Nuovo Cimento B (11), 67 (1982), 277.
doi: 10.1007/BF02721167. |
[4] |
R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems,, Comm. Math. Phys., 150 (1992), 289.
doi: 10.1007/BF02096662. |
[5] |
R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions,, Discrete Contin. Dynam. Systems, 5 (1999), 157.
|
[6] |
C. Fefferman, Interpolation and extrapolation of smooth functions by linear operators,, Rev. Mat. Iberoam., 21 (2005), 313.
|
[7] |
C. Fefferman, The structure of linear extension operators for $C^m$,, Rev. Mat. Iberoam., 23 (2007), 269.
|
[8] |
C. Fefferman, $C^m$ extension by linear operators,, Ann. of Math. (2), 166 (2007), 779.
doi: 10.4007/annals.2007.166.779. |
[9] |
C. Fefferman, Extension of $C^{m,\omega}$-smooth functions by linear operators,, Rev. Mat. Iberoam., 25 (2009), 1.
|
[10] |
C. Fefferman, Whitney's extension problems and interpolation of data,, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 207.
|
[11] |
G. Gallavotti, Perturbation theory for classical Hamiltonian systems,, in, 7 (1981), 359.
|
[12] |
L. Grafakos, "Classical and Modern Fourier Analysis,'', Pearson Education, (2004).
|
[13] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'', Encyclopedia of Mathematics and its Applications, 54 (1995).
|
[14] |
M. Nicol and A. Török, Whitney regularity for solutions to the coboundary equation on Cantor sets,, Math. Phys. Electron. J., 13 (2007).
|
[15] |
G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I. Birkhoff normal forms,, Ann. Henri Poincaré, 1 (2000), 223.
doi: 10.1007/PL00001004. |
[16] |
G. Popov, KAM theorem for Gevrey Hamiltonians,, Ergodic Theory Dynam. Systems, 24 (2004), 1753.
doi: 10.1017/S0143385704000458. |
[17] |
G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians,, Mat. Contemp., 26 (2004), 87.
|
[18] |
J. Pöschel, Integrability of Hamiltonian systems on Cantor sets,, Comm. Pure Appl. Math., 35 (1982), 653.
doi: 10.1002/cpa.3160350504. |
[19] |
J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems,, Math. Z., 202 (1989), 559.
doi: 10.1007/BF01221590. |
[20] |
M. Shub, "Global Stability of Dynamical Systems,'', Springer-Verlag, (1987).
|
[21] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'', Princeton Mathematical Series, 30 (1970).
|
[22] |
J. A. Vano, "A Nash-Moser Implicit Function Theorem with Whitney Regularity and Applications,'', Ph.D thesis, (2002), 02. Google Scholar |
[23] |
F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma,, Dyn. Syst., 18 (2003), 159.
doi: 10.1080/1468936031000117857. |
[24] |
H. Whitney, Analytic extensions of differentiable functions defined in closed sets,, Trans. Amer. Math. Soc., 36 (1934), 63.
|
[25] |
H. Whitney, Differentiable functions defined in arbitrary subsets of Euclidean space,, Trans. Amer. Math. Soc., 40 (1936), 309.
|
[26] |
J. Xu and J. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition,, J. Differential Equations, 235 (2007), 609.
doi: 10.1016/j.jde.2006.12.001. |
[27] |
X. Yuan, Construction of quasi-periodic breathers via KAM technique,, Comm. Math. Phys., 226 (2002), 61.
doi: 10.1007/s002200100593. |
[28] |
X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations,, J. Differential Equations, 230 (2006), 213.
doi: 10.1016/j.jde.2005.12.012. |
[29] |
E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I,, Comm. Pure Appl. Math., 28 (1975), 91.
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