doi: 10.3934/dcds.2021164
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Quantitative destruction of invariant circles

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

Received  March 2021 Revised  August 2021 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 & Continuous Dynamical Systems, doi: 10.3934/dcds.2021164
References:
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V. Bangert, Mather sets for twist maps and geodesics on tori, Dynamics Reported, 1 (1988), 1-45.   Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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

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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.  Google Scholar

[9]

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

[10]

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

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

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

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