March  2022, 42(3): 1569-1583. doi: 10.3934/dcds.2021164

Quantitative destruction of invariant circles

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Received  March 2021 Revised  August 2021 Published  March 2022 Early access  November 2021

For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency $ \omega $ of an integrable system by a trigonometric polynomial of degree $ N $ perturbation $ R_N $ with $ \|R_N\|_{C^r}<\epsilon $. We obtain a relation among $ N $, $ r $, $ \epsilon $ and the arithmetic property of $ \omega $, for which the area-preserving map admit no invariant circles with $ \omega $.

Citation: Lin Wang. Quantitative destruction of invariant circles. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1569-1583. doi: 10.3934/dcds.2021164
References:
[1]

V. Bangert, Mather sets for twist maps and geodesics on tori, Dynamics Reported, 1 (1988), 1-45. 

[2]

Q. Chen and C.-Q. Cheng, Regular dependence of the Peierls barriers on perturbations, J. Differential Equations, 262 (2017), 4700-4723.  doi: 10.1016/j.jde.2016.12.018.

[3]

C.-Q. Cheng and L. Wang, Destruction of Lagrangian torus in positive definite Hamiltonian systems, Geometric and Functional Analysis, 23 (2013), 848-866.  doi: 10.1007/s00039-013-0213-z.

[4]

G. Forni, Analytic destruction of invariant circles, Ergod. Th. & Dynam. Sys., 14 (1994), 267-298.  doi: 10.1017/S0143385700007872.

[5]

M.-R. Herman, Sur la conjugation diff$\acute{e}$rentiable des diff$\acute{e}$omorphismes du cercle $\grave{a}$ des rotations, Publ. Math. IHES, 49 (1979), 5-233.  doi: 10.1007/BF02684798.

[6]

M.-R. Herman, Sur Les Courbes Invariantes Par Les Diff$\acute{e}$omorphismes de L'anneau, Ast$\acute{\text{e}}$risque, 103-104 1983,221 pp.

[7]

M.-R. Herman, Sur Les Courbes Invariantes Par Les Diff$\acute{e}$omorphismes de L'anneau, Ast$\acute{\text{e}}$risque, 144 1986.

[8]

J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, ed. P. H. Rabinowitz et al. NATO ASI Series C, 209. Reidel: Dordrecht, (1987), 177–202.

[9]

J. N. Mather, Destruction of invariant circles, Ergod. Th. & Dynam. Sys., 8 (1988), 199-214.  doi: 10.1017/S0143385700009421.

[10]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.

[11]

W. M. Schmidt, Diophantine Approximation, Lecture Notes in Mathematics, 785. Springer, Berlin, 1980. x+299 pp.

[12]

L. Wang, Variational destruction of invariant circles, Discrete and Continuous Dynamical Systems-A, 32 (2012), 4429-4443.  doi: 10.3934/dcds.2012.32.4429.

[13]

L. Wang, Total destruction of Lagrangian tori, Journal of Mathematical Analysis and Applications, 410 (2014), 827-836.  doi: 10.1016/j.jmaa.2013.09.018.

[14]

L. Wang, Destruction of invariant circles for Gevrey area-preserving twist maps, J. Dynam. Differential Equations, 27 (2015), 283-295.  doi: 10.1007/s10884-014-9361-6.

[15]

A. Zygmund, Trigonometric Series, Third Edition Volumes Ⅰ & Ⅱ combined, with a foreword by Robert Fefferman. Cambridge University Press, Cambridge, 2002.

show all references

References:
[1]

V. Bangert, Mather sets for twist maps and geodesics on tori, Dynamics Reported, 1 (1988), 1-45. 

[2]

Q. Chen and C.-Q. Cheng, Regular dependence of the Peierls barriers on perturbations, J. Differential Equations, 262 (2017), 4700-4723.  doi: 10.1016/j.jde.2016.12.018.

[3]

C.-Q. Cheng and L. Wang, Destruction of Lagrangian torus in positive definite Hamiltonian systems, Geometric and Functional Analysis, 23 (2013), 848-866.  doi: 10.1007/s00039-013-0213-z.

[4]

G. Forni, Analytic destruction of invariant circles, Ergod. Th. & Dynam. Sys., 14 (1994), 267-298.  doi: 10.1017/S0143385700007872.

[5]

M.-R. Herman, Sur la conjugation diff$\acute{e}$rentiable des diff$\acute{e}$omorphismes du cercle $\grave{a}$ des rotations, Publ. Math. IHES, 49 (1979), 5-233.  doi: 10.1007/BF02684798.

[6]

M.-R. Herman, Sur Les Courbes Invariantes Par Les Diff$\acute{e}$omorphismes de L'anneau, Ast$\acute{\text{e}}$risque, 103-104 1983,221 pp.

[7]

M.-R. Herman, Sur Les Courbes Invariantes Par Les Diff$\acute{e}$omorphismes de L'anneau, Ast$\acute{\text{e}}$risque, 144 1986.

[8]

J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, ed. P. H. Rabinowitz et al. NATO ASI Series C, 209. Reidel: Dordrecht, (1987), 177–202.

[9]

J. N. Mather, Destruction of invariant circles, Ergod. Th. & Dynam. Sys., 8 (1988), 199-214.  doi: 10.1017/S0143385700009421.

[10]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.

[11]

W. M. Schmidt, Diophantine Approximation, Lecture Notes in Mathematics, 785. Springer, Berlin, 1980. x+299 pp.

[12]

L. Wang, Variational destruction of invariant circles, Discrete and Continuous Dynamical Systems-A, 32 (2012), 4429-4443.  doi: 10.3934/dcds.2012.32.4429.

[13]

L. Wang, Total destruction of Lagrangian tori, Journal of Mathematical Analysis and Applications, 410 (2014), 827-836.  doi: 10.1016/j.jmaa.2013.09.018.

[14]

L. Wang, Destruction of invariant circles for Gevrey area-preserving twist maps, J. Dynam. Differential Equations, 27 (2015), 283-295.  doi: 10.1007/s10884-014-9361-6.

[15]

A. Zygmund, Trigonometric Series, Third Edition Volumes Ⅰ & Ⅱ combined, with a foreword by Robert Fefferman. Cambridge University Press, Cambridge, 2002.

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